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Can you prove whether this Polish Olympiad puzzle is possible or not? :)

0 Comments

2024/11/02
11:03 UTC

A neat little problem, solution and extensions I made for a student.

0 Comments

2024/11/01
06:55 UTC

Need Android Testers for Math Exercise App

Hi Everyone
I need tester for my Entusia Math Exercise Application
All helps appreciated please test my app each day and if you can also follow our social media links that would be awesome.
Let me know if anything needed

Join Google Group First
https://groups.google.com/g/entusia-tester/

Then Download From Play Store
https://play.google.com/store/apps/details?id=com.entusia.entusia

Web Link
https://play.google.com/apps/testing/com.entusia.entusia

Reddit Community
https://www.reddit.com/r/Entusia/

Instagram
https://www.instagram.com/entusia_app

Discord
https://discord.gg/BW4Xg7nDdU

0 Comments

2024/10/30
12:16 UTC

Can you solve it? May fail !

1 Comment

2024/10/29
14:33 UTC

That is 2.6m tall casts a shadow that is 1.13 m long. at the same time,a nearby building cadts a shadow that is 46.75 m long. how tall is the building

1 Comment

2024/10/28
02:11 UTC

The most direct driving route between Disneyland and Universal Studios Hollywood is 35 miles. What is the average speed of a traveler if it takes 1 hour and 15 minutes to make this drive

3 Comments

2024/10/28
01:00 UTC

How long until the number of seconds since 0 AD equals 100000000000000?

This one popped into my head very randomly. So I asked an AI and am still waiting on an answer. Perhaps Reddit will be faster. I'll write a program, in bash first then C++ if that takes to long, when I have the time.

So this question will assume that 0 AD equals 0000/01/01 00:00:00, time zone agnostic, in the format YYYY/MM/DD HH:MM:SS, a year is 365 days, and a day is 24 hours. All to make it as simple as possible. I don't care that the Gregorian calendar has changed, or that the first year was 0001, or for sidereal days for that matter. I will include leap years just because but not leap days.

So the question is based on the number of characters in the format YYYYMMDDHHMMSS, 14, vs the number of characters in the seconds since 0 AD. As of 2024/10/26 21:27:30 that would be 61,236,059,250 seconds or 11 characters. When will the number of characters in the seconds since 0 AD be longer than the number of characters in the full date and time?

In thinking on it I first did a manual calculation for the date 10000/01/01 24:00:00 to see where the number of seconds was. I was disappointed and found it to be 315,360,086,400 or only 12 characters long to the formats now 14 characters. Well I'll let the AI make its attempt, it's only been a few hours so far, while I put together a program myself to run on my PC or maybe on my offices HPC if the AI takes way too long. Though I might need to try a different prompt first before I "waste" precious CPU time.

I would want to know that date and how long it is till then in years, days, hours, minutes and seconds. The time needed to calculate this would also be interesting if it really takes a long time. I don't think it would though.

Anyways how would you try this? I'm not great at math in any way beyond writing a program with basic functions. I'll drop my own program algorithm, in the "blob of text that reads like a flowchart" format, in the comments tomorrow. Till then I would love to hear of your own attempts or algorithms. I'll also drop the AIs time when I hear back or if I try a different prompt. I'll probably share those as well just because.

13 Comments

2024/10/27
04:21 UTC

Calculating the probability of 6 dice rolling at least one 6 (and other cases)

Hi, first of all, I'm no mathematician so maybe this is a dumb observation, but here I go.

I was watching a YouTube video about Pokémon games where you have a 1 in 8192 chance of encountering a "shiny" Pokémon (that is a Pokémon with different color than usual), and I started wondering what were the odds of encountering one if I were to trigger 8192 encounters, thinking it should be pretty high.

So I started calculating more simple cases, like the odds of rolling 6 dice and getting a six, or flipping two coins and getting heads at least once.

The case with the coins is really easy, there are 4 possible combinations and three of them have at least one heads, so 75% chance of success.

For the dice, I calculated the odds of not getting a 6 on any of them: (5/6)x(5/6)x(5/6)... so (5/6)^6, and then substracted it from 1, and got 66.51%, almost 2/3, thats way lower than I anticipated, but the interesting part is that now I had a general formula:

1 - ((x-1)/x)^x

If I roll 100 100-sided dice, the odds of getting at least one '100' are ~63.4%.

If I trigger 8192 encounters in Pokémon, the chances of one of them being a 'shiny' are ~63.21%.

Then I used an onine calculator to know the limit of that formula if x tends to infinity and I simply got:

1 - (1/e)

So if you roll an infinite number of infinite sided dice, your chances of rolling 'infinite' (or any other specific number) at least once are 1 - (1/e), or ~63.21%.

Also, if you want to know how many tries you need to get your desired result, you can just do:

1 - ((x-1)/x)^y (x being the possible outcomes and y being the number of tries)

So to know the chances of rolling a 6 with different numbers of dice you'd get:

1 die: 16.66%

6 dice: 66.51%

10 dice: 83.85%

15 dice: 93.51%

20 dice: 97.39%

4 Comments

2024/10/23
23:21 UTC

Rigor in Proofs...

Is there any "theorem" which was accepted, say in XV Century, and later when proofs became more rigourous was disproven?

2 Comments

2024/10/22
22:23 UTC

No one can solve this !

8 Comments

2024/10/22
20:59 UTC

How to Simplify

I am collecting different ways of simplifying a value or expression so far this is all what I got

combining like terms
reducing fractions into their lowest term
removing operation that doesn't change the value (ex: divided by one, multiplied by one, +0)

Do you guys have other activities or processes that simplifies an expression or value, that I can add into this list?

2 Comments

2024/10/20
09:58 UTC

How do I know when to use n or x in equations?

What can I do to determine if a number is a constant (n) or a variable (x)?

6 Comments

2024/10/20
01:42 UTC

Can Math Help Repair Democracy? | Sam Wang | TED - From detecting gerrymandered districts to predicting the impact of alternative election methods like ranked-choice voting, Sam Wang outlines how computer simulations can help fix the bugs in US democracy and make it more responsive to the people.

0 Comments

2024/10/14
10:59 UTC

I'm The Exoponential Asshole

We all know that the term exponential is used mostly wrongly these days. I can usually live with it, but... sometimes it's too much, and I can't stop myself from informing people.

Paraphrasing a recent example, "war drones can be produced exponentially now." No, dude, that's like letting drones mate... (Edit: to be more precise, like letting clone themselves.)That was a funny thought, but sadly I'm the funniest person at parties...

6 Comments

2024/10/13
23:10 UTC

Old 20

I've a 20.00 bill with 2 sets of double numbers, fancy serial numbers website said it was 92% cool but not what it's worth....FOR what it's worth the check out at Walmart wouldn't accept 2 of my old 20s someone had to change them. I've actually 4 1977 20s but this was the only one that was cool

0 Comments

2024/10/13
02:25 UTC

Using Math to Uncover the Best Full House in Poker :)

0 Comments

2024/10/12
17:44 UTC

Need a tapering function with limited types of operations

I found one that is perfect for the mod I'm making in a game but the game doesn't allow exponentiation

f(x) = - 10^(-.002(x-1500)) +1000

any tapering function that has a section that is similar should work but I am limited to add/subtract/multiply/division/modulus/squareroot/round

anyone know a tapering function limited to those operations? It can't be piecemeal either.

0 Comments

2024/10/11
18:04 UTC

The divisibility rules of every number from 101 to 200 with alternate rules for the composite numbers that end in 1, 3, 7 or 9

Number	Rule	Examples
101	The difference between 10 times the last digit and the rest of the number is a multiple of 101	1,010 is a multiple of 101 because 101 - 0 x 10 = 101 and 101 is a multiple of 101
102	The number is a multiple of 2 and 51 at the same time	1,020 is a multiple of 102 because it ends in 0, which means it's a multiple of 2, and 102 - 0 x 5 = 102, which means it's a multiple of 51
103	The sum of 31 times the last digit and the rest of the number is a multiple of 103	1,030 is a multiple of 103 because 103 + 0 x 31 = 103 and 103 is a multiple of 103
104	The number is a multiple of 8 and 13 at the same time	1,040 is a multiple of 104 because it ends in 040, which means it's a multiple of 8, and 104 + 0 x 4 = 104, which means it's a multiple of 13
105	The number is a multiple of 5 and 21 at the same time	1,050 is a multiple of 105 because it ends in 0, which means it's a multiple of 5, and 105 - 0 x 2 = 105, which means it's a multiple of 21
106	The number is a multiple of 2 and 53 at the same time	1,060 is a multiple of 106 because it ends in 0, which means it's a multiple of 2, and 106 + 0 x 16 = 106, which means it's a multiple of 53
107	The difference between 32 times the last digit and the rest of the number is a multiple of 107	1,070 is a multiple of 107 because 107 - 0 x 32 = 107 and 107 is a multiple of 107
108	The number is a multiple of 4 and 27 at the same time	1,080 is a multiple of 108 because it ends in 80, which means it's a multiple of 4, and 108 - 0 x 8 = 108, which means it's a multiple of 27
109	The sum of 11 times the last digit and the rest of the number is a multiple of 109	1,090 is a multiple of 109 because 109 + 0 x 11 = 109 and 109 is a multiple of 109
110	The number is a multiple of 10 and 11 at the same time	1,100 is a multiple of 110 because it ends in 0, which means it's a multiple of 10, and 110 - 0 = 110, which means it's a multiple of 11
111	The difference between 11 times the last digit and the rest of the number is a multiple of 111	1,110 is a multiple of 111 because 111 - 0 x 11 = 111 and 111 is a multiple of 111
112	The number is a multiple of 7 and 16 at the same time	10,080 is a multiple of 112 because 1,008 - 0 x 2 = 1,008, which means it's a multiple of 7, and ends in 0,080, which means it's a multiple of 16
113	The sum of 34 times the last digit and the rest of the number is a multiple of 113	1,017 is a multiple of 113 because 101 + 7 x 34 = 339 and 339 is a multiple of 113
114	The number is a multiple of 2 and 57 at the same time	1,026 is a multiple of 114 because it ends in 6, which means it's a multiple of 2, and 102 - 6 x 17 = 0, which means it's a multiple of 57
115	The number is a multiple of 5 and 23 at the same time	1,035 is a multiple of 115 because it ends in 5, which means it's a multiple of 5, and 103 + 5 x 7 = 138, which means it's a multiple of 23
116	The number is a multiple of 4 and 29 at the same time	1,044 is a multiple of 116 because it ends in 44, which means it's a multiple of 4, and 104 + 4 x 3 = 116, which means it's a multiple of 29
117	The difference between 35 times the last digit and the rest of the number is a multiple of 117	1,053 is a multiple of 117 because 105 - 3 x 35 = 0 and 0 is a multiple of 117
118	The number is a multiple of 2 and 59 at the same time	1,062 is a multiple of 118 because it ends in 2, which means it's a multiple of 2, and 106 + 2 x 6 = 118, which means it's a multiple of 59
119	The sum of 12 times the last digit and the rest of the number is a multiple of 119	1,071 is a multiple of 119 because 107 + 1 x 12 = 119 and 119 is a multiple of 119
120	The number is a multiple of 3 and 40 at the same time	1,080 is a multiple of 120 because 1 + 8 = 9, which means it's a multiple of 3, and ends in 080, which means it's a multiple of 40
121	The difference between 12 times the last digit and the rest of the number is a multiple of 121	1,089 is a multiple of 121 because 108 - 9 x 12 = 0 and 0 is a multiple of 121
122	The number is a multiple of 2 and 61 at the same time	1,098 is a multiple of 122 because it ends in 8, which means it's a multiple of 2, and 109 - 8 x 6 = 61, which means it's a multiple of 61
123	The sum of 37 times the last digit and the rest of the number is a multiple of 123	1,107 is a multiple of 123 because 110 + 7 x 37 = 369 and 369 is a multiple of 123
124	The number is a multiple of 4 and 31 at the same time	1,116 is a multiple of 124 because it ends in 16, which means it's a multiple of 4, and 111 - 6 x 3 = 93, which means it's a multiple of 31
125	The number ends in 000, 125, 250, 375, 500, 625, 750 or 875	1,000 is a multiple of 125 because it ends in 000
126	The number is a multiple of 2 and 63 at the same time	1,008 is a multiple of 126 because it ends in 8, which means it's a multiple of 2, and 100 + 8 x 19 = 252, which means it's a multiple of 63
127	The difference between 38 times the last digit and the rest of the number is a multiple of 127	1,016 is a multiple of 127 because 6 x 38 - 101 = 127 and 127 is a multiple of 127
128	The last 7 digits are a multiple of 128; the 1,000,000s digit is even and the last 6 digits are a multiple of 128 or the 1,000,000s digit is odd and the last 6 digits are 64 times an odd number	10,000,000 is a multiple of 128 because the 1,000,000s digit is 0, an even number, and the last 6 digits are 000,000, a multiple of 128
129	The sum of 13 times the last digit and the rest of the number is a multiple of 129	1,032 is a multiple of 129 because 103 + 2 x 13 = 129 and 129 is a multiple of 129
130	The number is a multiple of 10 and 13 at the same time	1,040 is a multiple of 130 because it ends in 0, which means it's a multiple of 10, and 104 + 0 x 4 = 104, which means it's a multiple of 13
131	The difference between 13 times the last digit and the rest of the number is a multiple of 131	1,048 is a multiple of 131 because 104 - 8 x 13 = 0 and 0 is a multiple of 131
132	The number is a multiple of 4 and 33 at the same time	1,056 is a multiple of 132 because it ends in 56, which means it's a multiple of 4, and 105 + 6 x 10 = 165, which means it's a multiple of 33
133	The sum of 40 times the last digit and the rest of the number is a multiple of 133	1,064 is a multiple of 133 because 106 + 4 x 40 = 266 and 266 is a multiple of 133
134	The number is a multiple of 2 and 67 at the same time	1,072 is a multiple of 134 because it ends in 2, which means it's a multiple of 2, and 107 - 2 x 20 = 67, which means it's a multiple of 67
135	The number is a multiple of 5 and 27 at the same time	1,080 is a multiple of 135 because it ends in 0, which means it's a multiple of 5, and 108 - 0 x 8 = 108, which means it's a multiple of 27
136	The number is a multiple of 8 and 17 at the same time	1,088 is a multiple of 136 because it ends in 088, which means it's a multiple of 8 and 108 - 8 x 5 = 68, which means it's a multiple of 17
137	The difference between 41 times the last digit and the rest of the number is a multiple of 137	1,096 is a multiple of 137 because 6 x 41 - 109 = 137 and 137 is a multiple of 137
138	The number is a multiple of 2 and 69 at the same time	1,104 is a multiple of 138 because it ends in 4, which means it's a multiple of 2, and 110 + 4 x 7 = 138, which means it's a multiple of 69
139	The sum of 14 times the last digit and the rest of the number is a multiple of 139	1,112 is a multiple of 139 because 111 + 2 x 14 = 139 and 139 is a multiple of 139
140	The number is a multiple of 7 and 20 at the same time	1,120 is a multiple of 140 because 112 - 0 x 2 = 112, which means it's a multiple of 7, and ends in 20, which means it's a multiple of 20
141	The difference between 14 times the last digit and the rest of the number is a multiple of 141	1,128 is a multiple of 141 because 112 - 8 x 14 = 0 and 0 is a multiple of 141
142	The number is a multiple of 2 and 71 at the same time	1,136 is a multiple of 142 because it ends in 6, which means it's a multiple of 2, and 113 - 6 x 7 = 71, which means it's a multiple of 71
143	The sum of 43 times the last digit and the rest of the number is a multiple of 143	1,001 is a multiple of 143 because 100 + 1 x 43 = 143 and 143 is a multiple of 143
144	The number is a multiple of 9 and 16 at the same time	10,080 is a multiple of 144 because 1 + 8 = 9, which means it's a multiple of 9, and ends in 0,080, which means it's a multiple of 16
145	The number is a multiple of 5 and 29 at the same time	1,015 is a multiple of 145 because it ends in 5, which means it's a multiple of 5, and 101 + 5 x 3 = 116, which means it's a multiple of 29
146	The number is a multiple of 2 and 73 at the same time	1,022 is a multiple of 146 because it ends in 2, which means it's a multiple of 2, and 102 + 2 x 22 = 146, which means it's a multiple of 73
147	The difference between 44 times the last digit and the rest of the number is a multiple of 147	1,029 is a multiple of 147 because 9 x 44 - 102 = 294 and 294 is a multiple of 147
148	The number is a multiple of 4 and 37 at the same time	1,036 is a multiple of 148 because it ends in 36, which means it's a multiple of 4, and 103 - 6 x 11 = 37, which means it's a multiple of 37
149	The sum of 15 times the last digit and the rest of the number is a multiple of 149	1,043 is a multiple of 149 because 104 + 3 x 15 = 149 and 149 is a multiple of 149
150	The number is a multiple of 3 and 50 at the same time	1,050 is a multiple of 150 because 1 + 5 = 6, which means it's a multiple of 3, and ends in 50, which means it's a multiple of 50
151	The difference between 15 times the last digit and the rest of the number is a multiple of 151	1,057 is a multiple of 151 because 105 - 7 x 15 = 0 and 0 is a multiple of 151
152	The number is a multiple of 8 and 19 at the same time	1,064 is a multiple of 152 because it ends in 064, which means it's a multiple of 8, and 106 + 4 x 2 = 114, which means it's a multiple of 19
153	The sum of 46 times the last digit and the rest of the number is a multiple of 153	1,071 is a multiple of 153 because 107 + 1 x 46 = 153 and 153 is a multiple of 153
154	The number is a multiple of 2 and 77 at the same time	1,078 is a multiple of 154 because it ends in 8, which means it's a multiple of 2, and 8 x 23 - 107 = 77, which means it's a multiple of 77
155	The number is a multiple of 5 and 31 at the same time	1,085 is a multiple of 155 because it ends in 5, which means it's a multiple of 5, and 108 - 5 x 3 = 93, which means it's a multiple of 31
156	The number is a multiple of 4 and 39 at the same time	1,092 is a multiple of 156 because it ends in 92, which means it's a multiple of 4, and 109 + 2 x 4 = 117, which means it's a multiple of 39
157	The difference between 47 times the last digit and the rest of the number is a multiple of 157	1,099 is a multiple of 157 because 9 x 47 - 109 = 314 and 314 is a multiple of 157
158	The number is a multiple of 2 and 79 at the same time	1,106 is a multiple of 158 because it ends in 6, which means it's a multiple of 2, and 110 + 6 x 8 = 158, which means it's a multiple of 79
159	The sum of 16 times the last digit and the rest of the number is a multiple of 159	1,113 is a multiple of 159 because 111 + 3 x 16 = 159 and 159 is a multiple of 159
160	The last 5 digits are a multiple of 160; the 10,000s digit is even and the last 4 digits are a multiple of 160 or the 10,000s digit is odd and the last 4 digits are 80 times an odd number	100,000 is a multiple of 160 because the 10,000s digit is 0, an even number, and the last 4 digits are 0,000, a multiple of 160
161	The difference between 16 times the last digit and the rest of the number is a multiple of 161	1,127 is a multiple of 161 because 112 - 7 x 16 = 0 and 0 is a multiple of 161
162	The number is a multiple of 2 and 81 at the same time	1,134 is a multiple of 162 because it ends in 4, which means it's a multiple of 2, and 113 - 4 x 8 = 81, which means it's a multiple of 81
163	The sum of 49 times the last digit and the rest of the number is a multiple of 163	1,141 is a multiple of 163 because 114 + 1 x 49 = 163 and 163 is a multiple of 163
164	The number is a multiple of 4 and 41 at the same time	1,148 is a multiple of 164 because it ends in 64, which means it's a multiple of 4, and 114 - 8 x 4 - 82, which means it's a multiple of 41
165	The number is a multiple of 5 and 33 at the same time	1,155 is a multiple of 165 because it ends in 5, which means it's a multiple of 5, and 115 + 5 x 10 = 165, which means it's a multiple of 33
166	The number is a multiple of 2 and 83 at the same time	1,162 is a multiple of 166 because it ends in 2, which means it's a multiple of 2, and 116 + 2 x 25 = 166, which means it's a multiple of 83
167	The difference between 50 times the last digit and the rest of the number is a multiple of 167	1,002 is a multiple of 167 because 100 - 2 x 50 = 0 and 0 is a multiple of 167
168	The number is a multiple of 8 and 21 at the same time	1,008 is a multiple of 168 because it ends in 008, which means it's a multiple of 8, and 100 - 8 x 2 = 84, which means it's a multiple of 21
169	The sum of 17 times the last digit and the rest of the number is a multiple of 169	1,014 is a multiple of 169 because 101 + 4 x 17 = 169 and 169 is a multiple of 169
170	The number is a multiple of 10 and 17 at the same time	1,020 is a multiple of 170 because it ends in 0, which means it's a multiple of 10, and 102 - 0 x 5 = 102, which means it's a multiple of 17
171	The difference between 17 times the last digit and the rest of the number is a multiple of 171	1,026 is a multiple of 171 because 102 - 6 x 17 = 0 and 0 is a multiple of 171
172	The number is a multiple of 4 and 43 at the same time	1,032 is a multiple of 172 because it ends in 32, which means it's a multiple of 4, and 103 + 2 x 13 = 129, which means it's a multiple of 43
173	The sum of 52 times the last digit and the rest of the number is a multiple of 173	1,038 is a multiple of 173 because 103 + 8 x 52 = 519 and 519 is a multiple of 173
174	The number is a multiple of 2 and 87 at the same time	1,044 is a multiple of 174 because it ends in 4, which means it's a multiple of 2, and 104 - 4 x 26 = 0, which means it's a multiple of 87
175	The number is a multiple of 7 and 25 at the same time	1,050 is a multiple of 175 because 105 - 0 x 2 = 105, which means it's a multiple of 7, and ends in 50, which means it's a multiple of 25
176	The number is a multiple of 11 and 16 at the same time	10,032 is a multiple of 176 because 1,003 - 2 = 1,001, which means it's a multiple of 11, and ends in 0,032, which means it's a multiple of 16
177	The difference between 53 times the last digit and the rest of the number is a multiple of 177	1,062 is a multiple of 177 because 106 - 2 x 53 = 0 and 0 is a multiple of 177
178	The number is a multiple of 2 and 89 at the same time	1,068 is a multiple of 178 because it ends in 8, which means it's a multiple of 2, and 106 + 8 x 9 = 178, which means it's a multiple of 89
179	The sum of 18 times the last digit and the rest of the number is a multiple of 179	1,074 is a multiple of 179 because 107 + 4 x 18 = 179 and 179 is a multiple of 179
180	The number is a multiple of 9 and 20 at the same time	1,080 is a multiple of 180 because 1 + 8 = 9, which means it's a multiple of 9, and ends in 80, which means it's a multiple of 20
181	The difference between 18 times the last digit and the rest of the number is a multiple of 181	1,086 is a multiple of 181 because 108 - 6 x 18 = 0 and 0 is a multiple of 181
182	The number is a multiple of 2 and 91 at the same time	1,092 is a multiple of 182 because it ends in 2, which means it's a multiple of 2, and 109 - 9 x 2 = 91, which means it's a multiple of 91
183	The sum of 55 times the last digit and the rest of the number is a multiple of 183	1,098 is a multiple of 183 because 109 + 8 x 55 = 549 and 549 is a multiple of 183
184	The number is a multiple of 8 and 23 at the same time	1,104 is a multiple of 184 because it ends in 104, which means it's a multiple of 8, and 110 + 4 x 7 = 138, which means it's a multiple of 23
185	The number is a multiple of 5 and 37 at the same time	1,110 is a multiple of 185 because it ends in 0, which means it's a multiple of 5, and 111 - 0 x 11 = 111, which means it's a multiple of 37
186	The number is a multiple of 2 and 93 at the same time	1,116 is a multiple of 186 because it ends in 6, which means it's a multiple of 2, and 111 + 6 x 28 = 279, which means it's a multiple of 93
187	The difference between 56 times the last digit and the rest of the number is a multiple of 187	1,122 is a multiple of 187 because 112 - 2 x 56 = 0 and 0 is a multiple of 187
188	The number is a multiple of 4 and 47 at the same time	1,128 is a multiple of 188 because it ends in 28, which means it's a multiple of 4, and 112 - 8 x 14 = 0, which means it's a multiple of 47
189	The sum of 19 times the last digit and the rest of the number is a multiple of 189	1,134 is a multiple of 189 because 113 + 4 x 19 = 189 and 189 is a multiple of 189
190	The number is a multiple of 10 and 19 at the same time	1,140 is a multiple of 190 because it ends in 0, which means it's a multiple of 10, and 114 + 0 x 2 = 114, which means it's a multiple of 19
191	The difference between 19 times the last digit and the rest of the number is a multiple of 191	1,146 is a multiple of 191 because 114 - 6 x 19 = 0 and 0 is a multiple of 191
192	The number is a multiple of 3 and 64 at the same time	1,000,128 is a multiple of 192 because 1 + 1 + 2 + 8 = 12, which means it's a multiple of 3, and ends in 000,128, which means it's a multiple of 64
193	The sum of 58 times the last digit and the rest of the number is a multiple of 193	1,158 is a multiple of 193 because 115 + 8 x 58 = 579 and 579 is a multiple of 193
194	The number is a multiple of 2 and 97 at the same time	1,164 is a multiple of 194 because it ends in 4, which means it's a multiple of 2, and 116 - 4 x 29 = 0, which means it's a multiple of 97
195	The number is a multiple of 5 and 39 at the same time	1,170 is a multiple of 195 because it ends in 0, which means it's a multiple of 5, and 117 + 0 x 4 = 117, which means it's a multiple of 39
196	The number is a multiple of 4 and 49 at the same time	1,176 is a multiple of 196 because it ends in 76, which means it's a multiple of 4, and 117 + 6 x 5 = 147, which means it's a multiple of 49
197	The difference between 59 times the last digit and the rest of the number is a multiple of 197	1,182 is a multiple of 197 because 118 - 2 x 59 = 0 and 0 is a multiple of 197
198	The number is a multiple of 2 and 99 at the same time	1,188 is a multiple of 198 because it ends in 8, which means it's a multiple of 2, and 118 + 8 x 10 = 198, which means it's a multiple of 99
199	The sum of 20 times the last digit and the rest of the number is a multiple of 199	1,194 is a multiple of 199 because 119 + 4 x 20 = 199 and 199 is a multiple of 199
200	The number ends in 000, 200, 400, 600 or 800	1,000 is a multiple of 200 because it ends in 000

0 Comments

2024/10/11
17:50 UTC

CompetifyHub October POTM

Hello, Competify Hub provides high quality problems monthly for this reddit server, we will provide the solution in the next month's post.

September POTM Solution: (√6)/2. Let f be the transformation that stretches the plane by a factor of OB/OA in the direction of OA, and let Q be the projection of P onto OA.

Also, let A’ = f(A), P’ = f(P), and Q’ = f(Q).

Note that under f, the ellipse becomes a circle with center O and radius OB, so 10∠P’OA’ = (360°)(1/6) = 60° because of the area condition.

Therefore, OA/OB = OA/OA’ = OQ/OQ’ = (cos 60°)/(cos 45°) = (√3)/(√2) = (√6)/2.

October POTM

Problem: In ∆ABC with AB = 13, BC = 14, and CA = 15, there is an ellipse inscribed in ∆ABC such that one focus is the orthocenter of ∆ABC. Find the length of the major axis of this ellipse as a common fraction.

0 Comments

2024/10/06
12:37 UTC

The divisibility rules of every number from 1 to 100 but this time with alternate methods for the composite numbers that end in 1, 3, 7 or 9

Number	Rule	Examples
1	Every number is a multiple of 1	10 is a multiple of 1
2	The number ends in 0, 2, 4, 6 or 8	10 is a multiple of 2 because it ends in 0
3	The sum of the digits is a multiple of 3	12 is a multiple of 3 because 1 + 2 = 3 and 3 is a multiple of 3
4	The last 2 digits are a multiple of 4; the 10s digit is even and the last digit is 0, 4 or 8, or the 10s digit is odd and the last digit is 2 or 6	100 is a multiple of 4 because the 10s digit is 0, an even number, and the last digit is 0
5	The number ends in 0 or 5	10 is a multiple of 5 because it ends in 0
6	The number is a multiple of 2 and 3 at the same time	12 is a multiple of 6 because it ends in 2, which means it's a multiple of 2, and 1 + 2 = 3, which means it's a multiple of 3
7	The difference between twice the last digit and the rest of the number is a multiple of 7	14 is a multiple of 7 because 4 x 2 - 1 = 7 and 7 is a multiple of 7
8	The last 3 digits are a multiple of 8; the 100s digit is even and the last 2 digits are a multiple of 8 or the 100s digit is odd and the last 2 digits are 4 times an odd number	1,000 is a multiple of 8 because the 100s digit is 0, an even number, and the last 2 digits are 00, a multiple of 8
9	The sum of the digits is a multiple of 9	18 is a multiple of 9 because 1 + 8 = 9 and 9 is a multiple of 9
10	The number ends in 0	100 is a multiple of 10 because it ends in 0
11	The difference between the last digit and the rest of the number is a multiple of 11	110 is a multiple of 11 because 11 - 0 = 11 and 11 is a multiple of 11
12	The number is a multiple of 3 and 4 at the same time	108 is a multiple of 12 because 1 + 8 = 9, which means it's a multiple of 3, and ends in 08, which means it's a multiple of 4
13	The sum of 4 times the last digit and the rest of the number is a multiple of 13	104 is a multiple of 13 because 10 + 4 x 4 = 26 and 26 is a multiple of 13
14	The number is a multiple of 2 and 7 at the same time	112 is a multiple of 14 because it ends in 2, which means it's a multiple of 2, and 11 - 2 x 2 = 7, which means it's a multiple of 7
15	The number is a multiple of 3 and 5 at the same time	105 is a multiple of 15 because 1 + 5 = 6, which means it's a multiple of 3, and ends in 5, which means it's a multiple of 5
16	The last 4 digits are a multiple of 16; the 1,000s digit is even and the last 3 digits are a multiple of 16 or the 1,000s digit is odd and the last 3 digits are 8 times an odd number	10,000 is a multiple of 16 because the 1,000s digit is 0, an even number, and the last 3 digits are 000, a multiple of 16
17	The difference between 5 times the last digit and the rest of the number is a multiple of 17	102 is a multiple of 17 because 10 - 2 x 5 = 0 and 0 is a multiple of 17
18	The number is a multiple of 2 and 9 at the same time	108 is a multiple of 18 because it ends in 8, which means it's a multiple of 2, and 1 + 8 = 9, which means it's a multiple of 9
19	The sum of twice the last digit and the rest of the number is a multiple of 19	114 is a multiple of 19 because 11 + 4 x 2 = 19 and 19 is a multiple of 19
20	The number ends in 00, 20, 40, 60 or 80	100 is a multiple of 20 because it ends in 00
21	The difference between twice the last digit and the rest of the number is a multiple of 21	105 is a multiple of 21 because 10 - 5 x 2 = 0 and 0 is a multiple of 21
22	The number is a multiple of 2 and 11 at the same time	110 is a multiple of 22 because it ends in 0, which means it's a multiple of 2, and 11 - 0 = 11, which means it's a multiple of 11
23	The sum of 7 times the last digit and the rest of the number is a multiple of 23	115 is a multiple of 23 because 11 + 5 x 7 = 46 and 46 is a multiple of 23
24	The number is a multiple of 3 and 8 at the same time	1,008 is a multiple of 24 because 1 + 8 = 9, which means it's a multiple of 3, and ends in 008, which means it's a multiple of 8
25	The number ends in 00, 25, 50 or 75	100 is a multiple of 25 because it ends in 00
26	The number is a multiple of 2 and 13 at the same time	104 is a multiple of 26 because it ends in 4, which means it's a multiple of 2, and 10 + 4 x 4 = 26, which means it's a multiple of 13
27	The difference between 8 times the last digit and the rest of the number is a multiple of 27	108 is a multiple of 27 because 8 x 8 - 10 = 54 and 54 is a multiple of 27
28	The number is a multiple of 4 and 7 at the same time	112 is a multiple of 28 because it ends in 12, which means it's a multiple of 4, and 11 - 2 x 2 = 7, which means it's a multiple of 7
29	The sum of thrice the last digit and the rest of the number is a multiple of 29	116 is a multiple of 29 because 11 + 6 x 3 = 29 and 29 is a multiple of 29
30	The number is a multiple of 3 and 10 at the same time	120 is a multiple of 30 because 1 + 2 = 3, which means it's a multiple of 3, and ends in 0, which means it's a multiple of 10
31	The difference between thrice the last digit and the rest of the number is a multiple of 31	124 is a multiple of 31 because 12 - 4 x 3 = 0 and 0 is a multiple of 31
32	The last 5 digits are a multiple of 32; the 10,000s digit is even and the last 4 digits are a multiple of 32 or the 10,000s digit is odd and the last 4 digits are 16 times an odd number	100,000 is a multiple of 32 because the 10,000s digit is 0, an even number, and the last 4 digits are 0,000, a multiple of 32
33	The sum of 10 times the last digit and the rest of the number is a multiple of 33	132 is a multiple of 33 because 13 + 2 x 10 = 33 and 33 is a multiple of 33
34	The number is a multiple of 2 and 17 at the same time	102 is a multiple of 34 because it ends in 2, which means it's a multiple of 2, and 10 - 2 x 5 = 0, which means it's a multiple of 17
35	The number is a multiple of 5 and 7 at the same time	105 is a multiple of 35 because it ends in 5, which means it's a multiple of 5, and 10 - 5 x 2 = 0, which means it's a multiple of 7
36	The number is a multiple of 4 and 9 at the same time	108 is a multiple of 36 because it ends in 08, which means it's a multiple of 4, and 1 + 8 = 9, which means it's a multiple of 9
37	The difference between 11 times the last digit and the rest of the number is a multiple of 37	111 is a multiple of 37 because 11 - 1 x 11 = 0 and 0 is a multiple of 37
38	The number is a multiple of 2 and 19 at the same time	114 is a multiple of 38 because it ends in 4, which means it's a multiple of 2, and 11 + 4 x 2 = 19, which means it's a multiple of 19
39	The sum of 4 times the last digit and the rest of the number is a multiple of 39	117 is a multiple of 39 because 11 + 7 x 4 = 39 and 39 is a multiple of 39
40	The last 3 digits are a multiple of 40; the 100s digit is even and the last 2 digits are 00, 40 or 80, or the 100s digit is odd and the last 2 digits are 20 or 60	1,000 is a multiple of 40 because the 100s digit is 0, an even number, and the last 2 digits are 00
41	The difference between 4 times the last digit and the rest of the number is a multiple of 41	123 is a multiple of 41 because 12 - 3 x 4 = 0 and 0 is a multiple of 41
42	The number is a multiple of 2 and 21 at the same time	126 is a multiple of 42 because it ends in 6, which means it's a multiple of 2, and 12 - 6 x 2 = 0, which means it's a multiple of 21
43	The sum of 13 times the last digit and the rest of the number is a multiple of 43	129 is a multiple of 43 because 12 + 9 x 13 = 129 and 129 is a multiple of 43
44	The number is a multiple of 4 and 11 at the same time	132 is a multiple of 44 because it ends in 32, which means it's a multiple of 4, and 13 - 2 = 11, which means it's a multiple of 11
45	The number is a multiple of 5 and 9 at the same time	135 is a multiple of 45 because it ends in 5, which means it's a multiple of 5, and 1 + 3 + 5 = 9, which means it's a multiple of 9
46	The number is a multiple of 2 and 23 at the same time	138 is a multiple of 46 because it ends in 8, which means it's a multiple of 2, and 13 + 8 x 7 = 69, which means it's a multiple of 23
47	The difference between 14 times the last digit and the rest of the number is a multiple of 47	141 is a multiple of 47 because 14 - 1 x 14 = 0 and 0 is a multiple of 47
48	The number is a multiple of 3 and 16 at the same time	10,032 is a multiple of 48 because 1 + 3 + 2 = 6, which means it's a multiple of 3, and ends in 0,032, which means it's a multiple of 16
49	The sum of 5 times the last digit and the rest of the number is a multiple of 49	147 is a multiple of 49 because 14 + 7 x 5 = 49 and 49 is a multiple of 49
50	The number ends in 00 or 50	100 is a multiple of 50 because it ends in 00
51	The difference between 5 times the last digit and the rest of the number is a multiple of 51	102 is a multiple of 51 because 10 - 2 x 5 = 0 and 0 is a multiple of 51
52	The number is a multiple of 4 and 13 at the same time	104 is a multiple of 52 because it ends in 04, which means it's a multiple of 4, and 10 + 4 x 4 = 26, which means it's a multiple of 13
53	The sum of 16 times the last digit and the rest of the number is a multiple of 53	106 is a multiple of 53 because 10 + 6 x 16 = 106 and 106 is a multiple of 53
54	The number is a multiple of 2 and 27 at the same time	108 is a multiple of 54 because it ends in 8, which means it's a multiple of 2, and 8 x 8 - 10 = 54, which means it's a multiple of 27
55	The number is a multiple of 5 and 11 at the same time	110 is a multiple of 55 because it ends in 0, which means it's a multiple of 5, and 11 - 0 = 11, which means it's a multiple of 11
56	The number is a multiple of 7 and 8 at the same time	1,008 is a multiple of 56 because 100 - 8 x 2 = 84, which means it's a multiple of 7, and ends in 008, which means it's a multiple of 8
57	The difference between 17 times the last digit and the rest of the number is a multiple of 57	114 is a multiple of 57 because 4 x 17 - 11 = 57 and 57 is a multiple of 57
58	The number is a multiple of 2 and 29 at the same time	116 is a multiple of 58 because it ends in 6, which means it's a multiple of 2, and 11 + 6 x 3 = 29, which means it's a multiple of 29
59	The sum of 6 times the last digit and the rest of the number is a multiple of 59	118 is a multiple of 59 because 11 + 8 x 6 = 59 and 59 is a multiple of 59
60	The number is a multiple of 3 and 20 at the same time	120 is a multiple of 60 because 1 + 2 = 3, which means it's a multiple of 3, and ends in 20, which means it's a multiple of 20
61	The difference between 6 times the last digit and the rest of the number is a multiple of 61	122 is a multiple of 61 because 12 - 2 x 6 = 0 and 0 is a multiple of 61
62	The number is a multiple of 2 and 31 at the same time	124 is a multiple of 62 because it ends in 4, which means it's a multiple of 2, and 12 - 4 x 3 = 0, which means it's a multiple of 31
63	The sum of 19 times the last digit and the rest of the number is a multiple of 63	126 is a multiple of 63 because 12 + 6 x 19 = 126 and 126 is a multiple of 63
64	The last 6 digits are a multiple of 64; the 100,000s digit is even and the last 5 digits are a multiple of 64 or the 100,000s digit is odd and the last 5 digits are 32 times an odd number	1,000,000 is a multiple of 64 because the 100,000s digit is 0, an even number, and the last 5 digits are 00,000, a multiple of 64
65	The number is a multiple of 5 and 13 at the same time	130 is a multiple of 65 because it ends in 0, which means it's a multiple of 5, and 13 + 0 x 4 = 13, which means it's a multiple of 13
66	The number is a multiple of 2 and 33 at the same time	132 is a multiple of 66 because it ends in 2, which means it's a multiple of 2, and 13 + 2 x 10 = 33, which means it's a multiple of 33
67	The difference between 20 times the last digit and the rest of the number is a multiple of 67	134 is a multiple of 67 because 4 x 20 - 13 = 67 and 67 is a multiple of 67
68	The number is a multiple of 4 and 17 at the same time	136 is a multiple of 68 because it ends in 36, which means it's a multiple of 4, and 6 x 5 - 13 = 17, which means it's a multiple of 17
69	The sum of 7 times the last digit and the rest of the number is a multiple of 69	138 is a multiple of 69 because 13 + 8 x 7 = 69 and 69 is a multiple of 69
70	The number is a multiple of 7 and 10 at the same time	140 is a multiple of 70 because 14 - 0 x 2 = 14, which means it's a multiple of 7, and ends in 0, which means it's a multiple of 10
71	The difference between 7 times the last digit and the rest of the number is a multiple of 71	142 is a multiple of 71 because 14 - 2 x 7 = 0 and 0 is a multiple of 71
72	The number is a multiple of 8 and 9 at the same time	1,008 is a multiple of 72 because it ends in 008, which means it's a multiple of 8, and 1 + 8 = 9, which means it's a multiple of 9
73	The sum of 22 times the last digit and the rest of the number is a multiple of 73	146 is a multiple of 73 because 14 + 6 x 22 = 146 and 146 is a multiple of 73
74	The number is a multiple of 2 and 37 at the same time	148 is a multiple of 74 because it ends in 8, which means it's a multiple of 2, and 8 x 11 - 14 = 74, which means it's a multiple of 37
75	The number is a multiple of 3 and 25 at the same time	150 is a multiple of 75 because 1 + 5 = 6, which means it's a multiple of 3, and ends in 50, which means it's a multiple of 25
76	The number is a multiple of 4 and 19 at the same time	152 is a multiple of 76 because it ends in 52, which means it's a multiple of 4, and 15 + 2 x 2 = 19, which means it's a multiple of 19
77	The difference between 23 times the last digit and the rest of the number is a multiple of 77	154 is a multiple of 77 because 4 x 23 - 15 = 77 and 77 is a multiple of 77
78	The number is a multiple of 2 and 39 at the same time	156 is a multiple of 78 because it ends in 6, which means it's a multiple of 2, and 15 + 6 x 4 = 39, which means it's a multiple of 39
79	The sum of 8 times the last digit and the rest of the number is a multiple of 79	158 is a multiple of 79 because 15 + 8 x 8 = 79 and 79 is a multiple of 79
80	The last 4 digits are a multiple of 80; the 1,000s digit is even and the last 3 digits are a multiple of 80 or the 1,000s digit is odd and the last 3 digits are 40 times an odd number	10,000 is a multiple of 80 because the 1,000s digit is 0, an even number, and the last 3 digits are 000, a multiple of 80
81	The difference between 8 times the last digit and the rest of the number is a multiple of 81	162 is a multiple of 81 because 16 - 2 x 8 = 0 and 0 is a multiple of 81
82	The number is a multiple of 2 and 41 at the same time	164 is a multiple of 82 because it ends in 4, which means it's a multiple of 2, and 16 - 4 x 4 = 0, which means it's a multiple of 41
83	The sum of 25 times the last digit and the rest of the number is a multiple of 83	166 is a multiple of 83 because 16 + 6 x 25 = 166 and 166 is a multiple of 83
84	The number is a multiple of 4 and 21 at the same time	168 is a multiple of 84 because it ends in 68, which means it's a multiple of 4, and 16 - 8 x 2 = 0, which means it's a multiple of 21
85	The number is a multiple of 5 and 17 at the same time	170 is a multiple of 85 because it ends in 0, which means it's a multiple of 5, and 17 - 0 x 5 = 17, which means it's a multiple of 17
86	The number is a multiple of 2 and 43 at the same time	172 is a multiple of 86 because it ends in 2, which means it's a multiple of 2, and 17 + 2 x 13 = 43, which means it's a multiple of 43
87	The difference between 26 times the last digit and the rest of the number is a multiple of 87	174 is a multiple of 87 because 4 x 26 - 17 = 87 and 87 is a multiple of 87
88	The number is a multiple of 8 and 11 at the same time	1,056 is a multiple of 88 because it ends in 056, which means it's a multiple of 8, and 105 - 6 = 99, which means it's a multiple of 11
89	The sum of 9 times the last digit and the rest of the number is a multiple of 89	178 is a multiple of 89 because 17 + 8 x 9 = 89 and 89 is a multiple of 89
90	The number is a multiple of 9 and 10 at the same time	180 is a multiple of 90 because 1 + 8 = 9, which means it's a multiple of 9, and ends in 0, which means it's a multiple of 10
91	The difference between 9 times the last digit and the rest of the number is a multiple of 91	182 is a multiple of 91 because 18 - 2 x 9 = 0 and 0 is a multiple of 91
92	The number is a multiple of 4 and 23 at the same time	184 is a multiple of 92 because it ends in 84, which means it's a multiple of 4, and 18 + 4 x 7 = 46, which means it's a multiple of 23
93	The sum of 28 times the last digit and the rest of the number is a multiple of 93	186 is a multiple of 93 because 18 + 6 x 28 = 186 and 186 is a multiple of 93
94	The number is a multiple of 2 and 47 at the same time	188 is a multiple of 94 because it ends in 8, which means it's a multiple of 2, and 8 x 14 - 18 = 94, which means it's a multiple of 47
95	The number is a multiple of 5 and 19 at the same time	190 is a multiple of 95 because it ends in 0, which means it's a multiple of 5, and 19 + 0 x 2 = 19, which means it's a multiple of 19
96	The number is a multiple of 3 and 32 at the same time	100,032 is a multiple of 96 because 1 + 3 + 2 = 6, which means it's a multiple of 3, and ends in 00,032, which means it's a multiple of 32
97	The difference between 29 times the last digit and the rest of the number is a multiple of 97	194 is a multiple of 97 because 4 x 29 - 19 = 97 and 97 is a multiple of 97
98	The number is a multiple of 2 and 49 at the same time	196 is a multiple of 98 because it ends in 6, which means it's a multiple of 2, and 19 + 6 x 5 = 49, which means it's a multiple of 49
99	The sum of 10 times the last digit and the rest of the number is a multiple of 99	198 is a multiple of 99 because 19 + 8 x 10 = 99 and 99 is a multiple of 99
100	The number ends in 00	1,000 is a multiple of 100 because it ends in 00

2 Comments

2024/10/04
16:31 UTC

This question has me stumped, im stuck between option b and c, which option is correct, and what would the others be? (for example, if c is correct, then explain how the elevator would move so that a is the correct answer)

https://preview.redd.it/nwemmc9lcerd1.png?width=687&format=png&auto=webp&s=ddf71d24faa7f20aad6dfe1ef82a14fb486134cc

7 Comments

2024/09/27
18:39 UTC