# Questions tagged [divisibility]

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

4,255 questions
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### Why does the statement “p is prime if it is divisible by only itself and 1” define only one prime number?

I'm having a bit of trouble understanding why this incorrect definition of primality only defines one prime number. "p is prime if it is divisible by only itself and 1." My understood definition of ...
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### Help to prove: $2^{(n+1)}+ 3^n+ 5^n$ is divisible by $6$ [on hold]

The question asks to use mathematical language to prove that: $2^{(n+1)}+ 3^n+ 5^n$ is divisible by $6$. such that n is any positive integer.
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### Can $ab+cd$ be an integer if $a, b, c, d$ are non integers and each 3 sum to an integer [on hold]

Can $ab+cd$ be an integer if $a, b, c, d$ are non integers and each 3 sum to an integer. This question in my opinion is tricky and I am unable to do much in it. However, after trying to substitute I ...
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### Prove that $(a^2b^2)^k + (b^2c^2)^k + (c^2a^2)^k$ is divisible for $\dfrac{1}{2}(a^4 + b^4 + c^4)$.

If $a$, $b$ and $c$ are a Pythagorean triple then prove that $(a^2b^2)^k + (b^2c^2)^k + (c^2a^2)^k$ is divisible for $\dfrac{1}{2}(a^4 + b^4 + c^4)$ for all integer $k \ge 2$. I cannot think of any ...
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### A problem of divisibility [on hold]

Say $n$ is a natural number such that $$3^n+3^{n+1}+...+3^{2n}=k^2$$ where $k$ is a natural number. Prove that $n$ is divisible by $4$.
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### Prove that the sum is not an integer

Prove that if a / b and c / d are two irreducible rational numbers such that gcd (b, d) = 1 then the sum (a/b + c/d) is not an integer. I was thinking about the proof by contradiction, but then I ...
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### How many five digit numbers formed from digits $1,2,3,4,5$ (used exactly once) are divisible by $12$?

How many five digit numbers formed from digits $1,2,3,4,5$ (used exactly once) are divisible by $12$? My answer is $24$ but I doubt if it's right or not. Sum of all the digits is $15$, so all the ...
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### Exercise on polynomial and GCD

Let $f(x), g(x)$ be relatively prime polynomial with coefficients in $\mathbb{Z}$. How can I prove that the GCD $(f(n),g(n)) = O(1)$ for $n \to \infty$, $n \in \mathbb{N}$? Thank you for the help!!
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### For positive integers prove that $a\Big|bc \implies a\Big|b \lor a\Big|c$

$a\Big | b,\; b = ak.$ $a\Big|c, c = al,$ So do I multiply $b$ and $c$ to get $a(kl)$ to prove that $bc = a$ multiplied by some integer $kl$ closed under multiplication?
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### prove: if $ab\mid cd$ and $a\mid c$ and $ab\nmid c$ then $b\mid d$ [closed]

I'm having a hard time proving the following claim: if $ab\mid cd$ and $a\mid c$ and $ab\nmid c$ then $b\mid d$ Any help would be appreciated
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### A006517: Numbers with $n\mid 2^n+2$

Problem 323 from the IMO 2009 reads: Prove that there are infinitely many positive integers n such that $2^n+2$ is divisible by $n$. An amazingly nice (and short) solution can be found here (see ...
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### Show that $a^n-b^n$ has a prime factor which does not divide $a-b$ for all $n>1$ .

I was asked to prove the following using the lifting the exponent lemma. Show that $a^n-b^n$ has a prime factor which does not divide $a-b$ for all $n>1$ . Using the first lemma, what I ...
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### Checking Proof for the following Divisibility

Prove that for every natural number $n\ge 2$, $n$ does not divide $n+1$. Proof: Suppose for every natural number $n\ge 2$, $n$ does divide $n+1.$ However, for natural numbers $a$ and $b,$ $a$ ...
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### Showing that, for $c,d\in\Bbb N$, $c|d$ implies $c\leq d$

I need help solving the following. My idea is to use Euclid's algorithm however I was told that I can simply prove this just with natural numbers. Prove that for all natural numbers $c$ and $d$, ...
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### number of solutions of $x^2=x$ divides the number of invertible elements in a ring

Let $A$ be a commutative ring with odd number of elements. If $n$ is the number of solutions of the equation $x^2=x,x\in A$, and $m$ is the number of invertible elements of $A$, prove that $n$ divides ...
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### Show that if $n$ and $k$ are positive integers, then $\lceil \frac{n}{k} \rceil = \lfloor \frac{n-1}{k} \rfloor + 1$

This is expanding on this question: Show that if $n$ and $k$ are positive integers, then $\lceil \frac{n}{k} \rceil = \lfloor \frac{n - 1}{k} \rfloor + 1$ as I'm unclear on how to solve this statement....
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