Question

Let $n$ be a positive integer which is not divisible by $2$ and $5. $ Prove that there is a multiple of $n$ consisting entirely of ones.

score 1.2k · Answer 1 · 2016-09-03 14:47:33

1.2k

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Let $n$ be a positive integer which is not divisible by $2$ and $5. $ Prove that there is a multiple of $n$ consisting entirely of ones.

Published on 03 Sep 2016 - 14:47

score 122 · Answer 2 · 2016-09-04 12:25:47

122

Views

Proof for $m!(n-m)!|n!$ for $m\leq n$

Published on 04 Sep 2016 - 12:25

score 263 · Answer 3 · 2026-03-26 07:59:15

263

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What is the smallest solution of the congruence $a^{x} \equiv 1 (\textrm{mod}\ n)$?

Published on 26 Mar 2026 - 7:59

score 1.3k · Answer 4 · 2026-03-31 08:10:51

1.3k

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Easy proof: $\binom{n}{k}$ is divisible by $n$ for $\gcd(n,k)=1$

Published on 31 Mar 2026 - 8:10

score 29 · Answer 5 · 2026-03-27 00:02:32

29

Views

Does one congruence imply the other?

Published on 27 Mar 2026 - 0:02

score 98 · Answer 6 · 2016-09-05 18:11:36

98

Views

Use mathematical induction to prove that $ \frac{(2n)!}{(2^n)} $ is a natural number for all n $\ge $ 0.

Published on 05 Sep 2016 - 18:11

score 116 · Answer 7 · 2016-09-05 19:35:53

116

Views

If $a\leq b$ then $bc+1$ does not divide $ab$.

Published on 05 Sep 2016 - 19:35

score 6.8k · Answer 8 · 2016-09-05 19:40:45

6.8k

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Are there integers $a,b,c$ such that $a$ divides $bc$, but $a$ does not divide $b$ and $a$ does not divide $c$?

Published on 05 Sep 2016 - 19:40

score 1.6k · Answer 9 · 2026-03-30 01:32:36

1.6k

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If $p$ is prime, show that $p\mid a^2 \implies p\mid a$

Published on 30 Mar 2026 - 1:32

score 109 · Answer 10 · 2016-09-06 17:59:08

109

Views

Prove that there is a $m$ that $f(m)=m$

Published on 06 Sep 2016 - 17:59

score 1.4k · Answer 11 · 2026-03-29 11:06:50

1.4k

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Proving the converse of an IMO problem: are there infinitely many pairs of positive integers (m,n) such that m divides n^2 + 1 and n divides m^2 + 1?

Published on 29 Mar 2026 - 11:06

score 534 · Answer 12 · 2026-03-30 23:20:46

534

Views

An interesting problem which only needs elementary number theory

Published on 30 Mar 2026 - 23:20

score 158 · Answer 13 · 2026-03-26 07:39:14

158

Views

On a conjectured relationship between the least prime factor and the Euler prime of an odd perfect number

Published on 26 Mar 2026 - 7:39

score 70 · Answer 14 · 2016-09-08 13:36:41

70

Views

Neat Proof for the statement: $\forall n\in\mathbb N.\ (n-1)(n+2) \not\equiv 2 \ \ \text{(mod 7)}$

Published on 08 Sep 2016 - 13:36

score 379 · Answer 15 · 2026-03-25 19:06:48

379

Views

least prime divisor of all numbers up to N=$100004$

Published on 25 Mar 2026 - 19:06

List Question

Let $n$ be a positive integer which is not divisible by $2$ and $5. $ Prove that there is a multiple of $n$ consisting entirely of ones.

Proof for $m!(n-m)!|n!$ for $m\leq n$

What is the smallest solution of the congruence $a^{x} \equiv 1 (\textrm{mod}\ n)$?

Easy proof: $\binom{n}{k}$ is divisible by $n$ for $\gcd(n,k)=1$

Does one congruence imply the other?

Use mathematical induction to prove that $ \frac{(2n)!}{(2^n)} $ is a natural number for all n $\ge $ 0.

If $a\leq b$ then $bc+1$ does not divide $ab$.

Are there integers $a,b,c$ such that $a$ divides $bc$, but $a$ does not divide $b$ and $a$ does not divide $c$?

If $p$ is prime, show that $p\mid a^2 \implies p\mid a$

Prove that there is a $m$ that $f(m)=m$

Proving the converse of an IMO problem: are there infinitely many pairs of positive integers (m,n) such that m divides n^2 + 1 and n divides m^2 + 1?

An interesting problem which only needs elementary number theory

On a conjectured relationship between the least prime factor and the Euler prime of an odd perfect number

Neat Proof for the statement: $\forall n\in\mathbb N.\ (n-1)(n+2) \not\equiv 2 \ \ \text{(mod 7)}$

least prime divisor of all numbers up to N=$100004$

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