Question

A positive integer has $1001$ digits all of which are $1$'s. When this number is divided by $1001$ find the remainder

score 1k · Answer 1 · 2026-02-23 08:34:15

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A positive integer has $1001$ digits all of which are $1$'s. When this number is divided by $1001$ find the remainder

Published on 23 Feb 2026 - 8:34

score 304 · Answer 2 · 2026-02-24 02:25:23

304

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Proving that 2 out of every 3 triangular numbers are divisible by 3

Published on 24 Feb 2026 - 2:25

score 141 · Answer 3 · 2020-07-06 21:08:45

141

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Show there are infinitely many natural numbers $n$ such that $n$ divides $16^n-1$

Published on 06 Jul 2020 - 21:08

score 81 · Answer 4 · 2020-07-08 00:49:17

81

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Proof of the Division Theorem (for all $a \in Z, b\in Z^+$, there exists $q,r\in Z$ such that $a=bq+r$ and $0\leq r\lt b$)

Published on 08 Jul 2020 - 0:49

score 294 · Answer 5 · 2026-04-02 01:38:28

294

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Divisibility Problem from an Olympiad Book

Published on 02 Apr 2026 - 1:38

score 171 · Answer 6 · 2026-03-29 16:01:42

171

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Find smallest $x$ such that $\gcd(a + x, b + x) = c$.

Published on 29 Mar 2026 - 16:01

score 1k · Answer 7 · 2020-07-09 12:40:04

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Given a positive integer $t$ does there always exist a natural number $k$ such that $(k!)^2$ is a factor of $(2k-t)!$?

Published on 09 Jul 2020 - 12:40

score 50 · Answer 8 · 2020-07-09 18:36:17

50

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Is this a valid proof for checking if a number K is dividable by 11,

Published on 09 Jul 2020 - 18:36

score 78 · Answer 9 · 2026-02-23 12:05:35

78

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If $ a-b \mid ax-by$, then $\gcd(x,y) \ne1$?

Published on 23 Feb 2026 - 12:05

score 263 · Answer 10 · 2026-03-31 16:26:56

263

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Mordell equation with prime-squared constant

Published on 31 Mar 2026 - 16:26

score 36 · Answer 11 · 2020-07-14 21:32:44

36

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Determining which members of a set are an integeral multiple of a member of a different set

Published on 14 Jul 2020 - 21:32

score 459 · Answer 12 · 2026-03-29 18:33:27

459

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On the diophantine equation $x^{m-1}(x+1)=y^{n-1}(y+1)$ with $x>y$, over integers greater or equal than two

Published on 29 Mar 2026 - 18:33

score 193 · Answer 13 · 2020-07-16 08:46:47

193

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Let $n$ be an odd positive integer. Show that $24 \vert(n^3-n).$

Published on 16 Jul 2020 - 8:46

score 555 · Answer 14 · 2026-03-28 15:21:18

555

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There is only one positive integer that is both the product and sum of all its proper positive divisors, and that number is $6$.

Published on 28 Mar 2026 - 15:21

score 484 · Answer 15 · 2020-07-17 08:55:04

484

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How to prove divisibility by $7$?

Published on 17 Jul 2020 - 8:55

List Question

A positive integer has $1001$ digits all of which are $1$'s. When this number is divided by $1001$ find the remainder

Proving that 2 out of every 3 triangular numbers are divisible by 3

Show there are infinitely many natural numbers $n$ such that $n$ divides $16^n-1$

Proof of the Division Theorem (for all $a \in Z, b\in Z^+$, there exists $q,r\in Z$ such that $a=bq+r$ and $0\leq r\lt b$)

Divisibility Problem from an Olympiad Book

Find smallest $x$ such that $\gcd(a + x, b + x) = c$.

Given a positive integer $t$ does there always exist a natural number $k$ such that $(k!)^2$ is a factor of $(2k-t)!$?

Is this a valid proof for checking if a number K is dividable by 11,

If $ a-b \mid ax-by$, then $\gcd(x,y) \ne1$?

Mordell equation with prime-squared constant

Determining which members of a set are an integeral multiple of a member of a different set

On the diophantine equation $x^{m-1}(x+1)=y^{n-1}(y+1)$ with $x>y$, over integers greater or equal than two

Let $n$ be an odd positive integer. Show that $24 \vert(n^3-n).$

There is only one positive integer that is both the product and sum of all its proper positive divisors, and that number is $6$.

How to prove divisibility by $7$?

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