List questions from divisibility

Published on 11 Apr 2026 - 16:21

30

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If :$\sum_{k=1}^{n}k^p =(\frac{n(n+1)}{2})\mod(p)$ how i deduce the remain of :$\sum_{k=1}^{n}k^{-p}$?

Published on 29 Mar 2026 - 19:14

43

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Tools for dealing with a divisibility problem with powers of 2 and 3?

Published on 12 Apr 2026 - 1:30

67

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When does $c\mid a(n+x)+b+1$, if we know that $c\mid an+b$?

Published on 09 Jun 2015 - 8:20

127

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Which prime divides $18^{29}+1$?

Published on 29 Mar 2026 - 19:09

57

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using Fibonacci numbers prove that if $d|n$ then $F_d|F_n$

Published on 09 Jun 2015 - 12:49

1.9k

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If $p$ is prime amd $1+\frac{1}{2}+\frac{1}{3}+\ldots + \frac{1}{p-1}=\frac{a}{b}$ then show that p divides a. Moreover if $p>3$ then $p^2$ divides a.

Published on 09 Jun 2015 - 13:47

10.4k

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Prove $\,\gcd(m,n)\operatorname{lcm}(m,n) = mn$

Published on 02 Apr 2026 - 21:39

520

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Is it true that $(pq,(p-1)(q-1)) =1 \iff (pq,\operatorname{lcm}(p-1,q-1))=1$?

Published on 11 Apr 2026 - 16:20

135

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Number of binomial coefficients , ${ n \choose k}$ k $\in$ [0,n] , that are divisible by a prime p?

Published on 10 Jun 2015 - 12:26

1.8k

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Show that $30 \mid (n^9 - n)$

Published on 10 Jun 2015 - 20:38

791

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Problem from Olympiad from book Arthur Engel

Published on 11 Apr 2026 - 2:26

110

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Does $p^n$ divide $\binom{p^{n+m-1}}{m}$?

Published on 11 Jun 2015 - 13:12

767

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Non existence of absolute euler pseudoprimes

Published on 26 Mar 2026 - 6:25

169

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When will $ax+1$ be divisible by $b$?

Published on 11 Apr 2026 - 8:21

List Question

First contest problem

If :$\sum_{k=1}^{n}k^p =(\frac{n(n+1)}{2})\mod(p)$ how i deduce the remain of :$\sum_{k=1}^{n}k^{-p}$?

Tools for dealing with a divisibility problem with powers of 2 and 3?

When does $c\mid a(n+x)+b+1$, if we know that $c\mid an+b$?

Which prime divides $18^{29}+1$?

using Fibonacci numbers prove that if $d|n$ then $F_d|F_n$

If $p$ is prime amd $1+\frac{1}{2}+\frac{1}{3}+\ldots + \frac{1}{p-1}=\frac{a}{b}$ then show that p divides a. Moreover if $p>3$ then $p^2$ divides a.

Prove $\,\gcd(m,n)\operatorname{lcm}(m,n) = mn$

Is it true that $(pq,(p-1)(q-1)) =1 \iff (pq,\operatorname{lcm}(p-1,q-1))=1$?

Number of binomial coefficients , ${ n \choose k}$ k $\in$ [0,n] , that are divisible by a prime p?

Show that $30 \mid (n^9 - n)$

Problem from Olympiad from book Arthur Engel

Does $p^n$ divide $\binom{p^{n+m-1}}{m}$?

Non existence of absolute euler pseudoprimes

When will $ax+1$ be divisible by $b$?

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