# Questions tagged [divisibility]

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

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### Question on binomial expansion - divisibility by power of divisor

By the Binomial theorem, one has that $$(a+b)^n = a^n + \binom{n}{1} \ a^{n-1}b + \binom{n}{2}\ a^{n-2}b^2 + \dots + b^n$$ Let us suppose that $d\mid (a+b)$, where $\mid$ means "divides", ...
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### Prove by mathematical induction, that $81\times 3^{2n} - 2^{2n}$ is divisible by $5$ where $n \in \mathbb{Z}^+$

For all $k$, the equation I came up with is $3^{4+2k} = 5m + 2^{2k}$ where $m$ is a positive integer. For all $k+1$, the expression is $3^{6+2k} - 2^{2k+2}$. I tried to plug in the first equation to ...
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### Prove the statements.

Prove that, f(x)=0 when k is not prime f(x)=1 when k is prime N.B: Here f(x)= a mod k means the reminder we get by dividing 'a' by 'k' I found that the statements are right by testing those. But I ...
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### Pythagorean triplets which satisfy the prime counting function $\omega(n)$

Let $\omega(n)$ be the number of prime factor of $n$. Out of the first $9.2 \times 10^9$ Pythagorean triplets $a^2 + b^2 = c^2$ (in ascending order of the hypotenuse) I found about $1.03 \times 10^7$ ...
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### Relationship between primes, right triangles and homogeneous polynomials

It is known that if $x^2 + y^2 = z^2$ is a primitive Pythagorean triplet then $z$ is not divisible by any prime of the form $4k-1$. The following is a generalization of this classical result which ...
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### If $\gcd(a,p)=1$, $p$ is prime, prove that there are $1\le x, y\le [\sqrt p]$ that $ax≡y\pmod p$ or $ax≡-y\pmod p$

If $\gcd(a,p)=1$, $p$ is prime, prove that there are $1\le x, y\le [\sqrt p]$ that $ax≡y\pmod p$ or $ax≡-y\pmod p$ I met this when looking proof of theory which says if $p=4k+1$ then there are $a,b$ ...
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### prove that for all $n≥2$ $n^4+4^n$ is not prime [duplicate]

Prove that $4^n+n^4$ is not prime for $n≥2$. I've found if $n≠5k$ then $5$ divides all equation. We get it from residues when divided by $5$. Also it’s clear that if $n=2k$ then equation is not prime ...
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### For how many natural numbers(<=100) is $1111^n +2222^n+3333^n+4444^n$ divisible by 10?

For how many natural numbers (0 not included) $n \leq 100$ is $1111^n +2222^n+3333^n+4444^n$ divisible by 10? I factored out $1111^n$ and got $1111^n(1+2^n+3^n+4^n)$. So $1+2^n+3^n+4^n$ must be ...
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### When we are finding a root to a polynomial say f(x), we put some constant

When we are finding a root to a polynomial say $f(x)$, we put some constant $a=x$ and check whether it is zero or not.Suppose we have successfully found a constant say $a$, then we say that $x-a=0$ is ...
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### Looking for divisibility by 29 and a general proof( if any) [closed]

Is there any elegant way to prove that 28C14 -1 is divisible by 29? Also, is this kind of a result a theorem or a generalisation? If so please do help... note i do mean 28 choose 14
### Let a and b be two positive integers with $d = \gcd(a,b)$. Show that $\gcd\left(2^a-1,2^b-1\right) = 2^d-1$. [duplicate]
Let a and b be two positive integers with $d = \gcd(a,b)$. Show that $\gcd\left(2^a-1,2^b-1\right) = 2^d-1$.
I believe that I intuitively understand that if $a$ divides $c$ and $b$ divides $c$ and if $a$ and $b$ are coprime, then their product $ab$ must also divide $c$. What would be a convincing proof of ...