List questions from divisibility

Published on 25 Sep 2016 - 9:09

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Proof that if $n>0$, $a=nm$, $b=np$, $c=nq$, then $\gcd(m,p,q)=1$ iff $n=\gcd(a,b,c)$

Published on 30 Mar 2026 - 3:56

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a divides bc if and only if

Published on 30 Mar 2026 - 3:54

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What day of the week will be $1993^3$ days from Sunday?

Published on 26 Sep 2016 - 4:05

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What is the remainder when 10,987,654 is divided by 2,103?

Published on 26 Sep 2016 - 4:22

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Show that the numerator of $1+\frac12 +\frac13 +\cdots +\frac1{96}$ is divisible by $97$

Published on 02 Apr 2026 - 18:42

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Proving a prime $p$ divides $n$ if and only if $p^2$ divides $n^2$

Published on 29 Sep 2016 - 0:50

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Greatest common denominator: $x\mid a$ and $x\mid b$ iff $x\mid\gcd(a,b)$

Published on 30 Mar 2026 - 3:55

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Prove that $q=4^n+1$ is prime iff $3^{\frac{(q-1)}2}=-1\bmod q$

Published on 02 Oct 2016 - 0:21

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Show that $p$ is a divisor of numbers $m^2 + 1$, $n^2 + 2$, if and only if it is a divisor of some number of the form $k^4 + 1$.

Published on 25 Mar 2026 - 12:50

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$\sum_{i=1}^{[ a/2]} [\frac {ib}{a} ] + \sum_{i=1}^{[ b/2]} [\frac {ja}{b} ] = [a/2][b/2] + [(a,b)/2].$

Published on 25 Mar 2026 - 11:14

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Why do some divisibility rules work only in base 10?

Published on 26 Mar 2026 - 17:29

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If $p$ is a prime number, prove that ${2p \choose p} - 2$ is divisible by $p^2$ without using Wolstenholme's theorem.

Published on 04 Oct 2016 - 4:07

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Divisibility Biconditional Proof

Published on 04 Oct 2016 - 13:41

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Find $\gcd(a,b)$ knowing that $(a^3+6b^2,6480)=216$ and $11a+8b \mid 540$

Published on 27 Mar 2026 - 3:43

List Question

Divisors problem

Proof that if $n>0$, $a=nm$, $b=np$, $c=nq$, then $\gcd(m,p,q)=1$ iff $n=\gcd(a,b,c)$

a divides bc if and only if

What day of the week will be $1993^3$ days from Sunday?

What is the remainder when 10,987,654 is divided by 2,103?

Show that the numerator of $1+\frac12 +\frac13 +\cdots +\frac1{96}$ is divisible by $97$

Proving a prime $p$ divides $n$ if and only if $p^2$ divides $n^2$

Greatest common denominator: $x\mid a$ and $x\mid b$ iff $x\mid\gcd(a,b)$

Prove that $q=4^n+1$ is prime iff $3^{\frac{(q-1)}2}=-1\bmod q$

Show that $p$ is a divisor of numbers $m^2 + 1$, $n^2 + 2$, if and only if it is a divisor of some number of the form $k^4 + 1$.

$\sum_{i=1}^{[ a/2]} [\frac {ib}{a} ] + \sum_{i=1}^{[ b/2]} [\frac {ja}{b} ] = [a/2][b/2] + [(a,b)/2].$

Why do some divisibility rules work only in base 10?

If $p$ is a prime number, prove that ${2p \choose p} - 2$ is divisible by $p^2$ without using Wolstenholme's theorem.

Divisibility Biconditional Proof

Find $\gcd(a,b)$ knowing that $(a^3+6b^2,6480)=216$ and $11a+8b \mid 540$

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