# Tagged Questions

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### How to find a modular multiplicative inverse when GCD is not 1

I am working on a problem that requires finding a multiplicative inverse of two numbers, but my algorithm is failing for a very simple reason: the GCD of the two numbers isn't 1. I figured I must've ...
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### Direct proof using modular arithmetic

Give a direct proof of $8\mid (3^n + 5^n)$ for all odd natural numbers. I know how to prove this by induction, I am not sure how to go about it using a direct proof. I would start by saying that ...
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### For every integer, some multiple of it is of the form $99 \ldots 900 \ldots 00$

The goal is to prove that for every positive integer $z$ there exists a positive integer $a$ such that $az = 99 \ldots 9900 \ldots 00$. Let $a = \frac {99 \ldots 9900 \ldots 00}{z}$ That ...
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### Solving $a^2+b^2\equiv 0$ mod $c$ for distinct integers $a,b,c$ constrained between two consecutive squares

Show that for any natural number $n$, between $n^2$ and $(n+1)^2$ one can find three distinct natural numbers $a,b,c$ such that $a^2+b^2$ is divisible by $c$. A friend and I found a general case that ...
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### Prove that a sequence of $11$ numbers always contains six numbers summing up to a multiple of $6$.

Prove that a sequence of $11$ numbers always contains six numbers summing up to a multiple of $6$. This is a problem from a selection to IMO 2014. I like thinking about this problem, it is ...
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### Proving that if $\;5\mid(a+11)\;$ and $\;5\mid(16-b),\;$ then $\;5\mid(a+b)$
Can you please help me a bit with this question? How do we show that $\;$ if $\;\;5\mid(a+11)\;$ and $\;5\mid(16-b),\;\;$ then $\;5\mid(a+b)\;$? Thanks a lot!