# 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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### Prove that following is always divisble by 5 [closed]

I am not getting how to prove that $1+ 2^{2^{4n-2}}$ is always divisible by 5 for every natural $n > 2$.
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### Proof that if $\gcd(a,p^2)=p$ and $\gcd(b,p^3)=p^2$ then $(ab,p^4)=p^3$ [duplicate]

So $p =ax+p^2y$ and $p^2 =bz+p^3w$ $ax= p-p^2y$ and $bz= p^2-p^3w$ $axbz= p^3-p^4w-p^4y+p^5wy$ $abxz= p^3-p^4(w+y-pwy)$ $abxz+p^4(w+y-pwy) = p^3$ How can I say that $\gcd(ab,p^4) = p^3$ ?
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### On solutions of $\varphi(n)=\frac{1}{2n}\sum_{1\leq d\mid n}\varphi(dn)$, where $\varphi(m)$ denotes the Euler's totient function

I wondered if one can to get easily an answer for the following question (I have thought about the other direction $\Leftarrow$). I don't know if it is in the literature, please refer it in comments ...
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### Find all the primes $p$, $q$ such that $pq|(5^p - 2^p)(5^q - 2^q)$.

First question for my typing mistake : Find all the primes $p$, $q$ such that $(5^p - 2^p)(5^q - 2^q)|pq$. I am extremely sorry. The question should be : Find all the primes $p$, $q$ such that ...
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### relation between concatenation and addition regarding modular arithmetic

This question turned out to be not so clear to me. When we concatenate two numbers (X and Y) the mod 23 of the new number XY is 0. The concatenation of (X-a) and (Y-b) , which is (X-a)(Y-b) also ...
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### Semiprimes Totient divisibility

please consider my question: Let $n$ be a positive semiprime, which is known and very very big. Let $\phi(n)$ its Euler Totient value. Consider that $n$ is big enough that factorization is unpractical ...
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### On integers $n\geq 1$ for which $n$ divides $\sum_{k=1}^n R_k$, where $R_k$ denotes the $k$-th Ramanujan prime

For integers $n\geq 1$ in this post we denote the Ramanujan primes as $R_n$, see for example the Wikipedia Ramanujan prime or . I don't know if my question is in the literature but I think that it ...
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### How can you prove the existence of an integer for ab congruent to 1 (mod p). [duplicate]

With a prime number p and a not equal to 0, how can I prove that there exists an integer b such that ab is congruent to 1 (mod p)? What I have done so far is I know that the GCD(p,a) is either 1 if p ...
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### Find all positive integers $n$ such that $n^2 \not\mid (n - 2)!$

Find all positive integers $n$ such that $n^2 \not\mid (n - 2)!$ We have that $\gcd(n - 1, n) = 1$, then $n^2 \not\mid (n - 1)!$ Of course, primes and Carmichael numbers are solutions to this ...
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### If a prime and its square both divide a number n, prove that $n=a^2 b^3$
Lets call a number $n$ a fortified number if $n>0$ and for every prime number $p$, if $p|n$ then $p^2|n$. Given a fortified number, prove that there exists $a,b$ such that $n=a^2b^3$. I know that ...
### If $p \ge 5$ is prime prove that $\sum_{i=1}^{p-2}\sum_{j=i+1}^{p-1} ij$ is divisible by $p$
If $p \ge 5$ is prime prove that $$\sum_{i=1}^{p-2}\sum_{j=i+1}^{p-1}ij$$ Attempt. We have \sum_{i=1}^{p-2}\sum_{j=i+1}^{p-1}ij = \sum_{i=1}^{p-2}i \left[ \frac{(p-1)p - i(i+1)}{2}\right] = \frac{(...