Questions tagged [divisibility]

This tag is for questions about divisibility, that is, determining when one thing is a multiple of another thing.

4,296 questions
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Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$

For all $a, m, n \in \mathbb{Z}^+$, $$\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$$
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Are half of all numbers odd?

Plato puts the following words in Socrates' mouth in the Phaedo dialogue: I mean, for instance, the number three, and there are many other examples. Take the case of three; do you not think it may ...
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How do you explain to a 5th grader why division by zero is meaningless?

I wish to explain my younger brother: he is interested and curious, but he cannot grasp the concepts of limits and integration just yet. What is the best mathematical way to justify not allowing ...
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Division by $0$ and its restrictions

Consider the following expression: $$\frac{1}{2} \div \frac{4}{x}$$ Over here, one would state the restriction as $x \neq 0$, as that would result in division by $0$. But if we rearrange the ...
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How to solve these two simultaneous “divisibilities” : $n+1\mid m^2+1$ and $m+1\mid n^2+1$

Is it possible to find all integers $m>0$ and $n>0$ such that $n+1\mid m^2+1$ and $m+1\,|\,n^2+1$ ? I succeed to prove there is an infinite number of solutions, but I cannot progress anymore. ...
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Any rectangular shape on a calculator numpad when divided by 11 gives an integer. Why?

I have come across this fact a very long time ago, but I still can't tell why is this happening. Given the standard calculators numpad: 7 8 9 4 5 6 1 2 3 if you ...
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What is the smallest positive multiple of 450 whose digits are all zeroes and ones?

What is the smallest positive multiple of 450 whose digits are all zeroes and ones? I tried guess and check but the numbers grew big fast. Thanks in advance!
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How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes? [duplicate]

Possible Duplicate: Highest power of a prime $p$ dividing $N!$ How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes?
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Why is $a^n - b^n$ divisible by $a-b$?

I did some mathematical induction problems on divisibility $9^n$ $-$ $2^n$ is divisible by 7. $4^n$ $-$ $1$ is divisible by 3. $9^n$ $-$ $4^n$ is divisible by 5. Can these be generalized as $a^n$ $-$...
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