# Tagged Questions

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

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### Demonstrate that $\int_0^1{\frac{(x^2+x+1)^{4n+1}- x}{x^2+1}dx}$ is a rational number

I thought about proving $x^2+1$ divides $(x^2+x+1)^{4n+1}- x$ , but I don't know how.
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### Which prime factors of $8^{8^8}+1$ are known?

We have the partial factorization $$8^{8^8}+1=(2^{2^{24}}+1)\cdot (2^{2^{25}}-2^{2^{24}}+1)$$ The first factor is $F_{24}$. It is composite, but no prime factor is known. A prime factor of the ...
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### GCD divisibility of LCM

Show that the following conditions are equivalent: i) There exist positive integers $a,b$ such that $\gcd(a,b)=d$ and $\operatorname{lcm}(a,b)=m$. ii) $dâˆ£m$ The first direction is very ...
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### For any $a$ in $\Bbb Z$, prove that $6|a(a+5)(a+10)$

So I am given this question for my number theory and proof class: For any $a \in \Bbb Z$, prove that $6|a(a+5)(a+10)$. I've thought about a few different ways to approach this. I think I could ...
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### How many numbers less than 100 have the sum of factors as odd?

How many numbers less than 100 have the sum of factors as odd? Answer is 16 This question and explanation is taken from careerbless.com The link given derives the answer using some properties ...
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### Help - remainders when number is divided

Please, give me hints, I've no idea ;): Find greatest number $x$ such that $x<1000$ and $x$ divided by $4$ gives remainder $3$, divided by $5$ gives remainder $4$, and divided by $6$ gives ...
Suppose that $n=x^2$ is a perfect square with an even number of base-10 digits. Assume that when n is written backwards, you get another perfect square $y^2$. Prove that 121|n. (Use the mod 11 ...
### Proof checking Number theory: prove that $d\nmid a^{2^{n}}+1$.
Let $a, d, n$ be positive integers with $2<d<2^{n+1}$, prove that $d\nmid a^{2^{n}}+1$. I've made some preliminary observations: I hypothesize that for any $n$, $a^{2^n}+1=2\prod p$ where the ...