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Prove that the multiplicative inverse of $a$ modulo $m$ exists if and only if $a$ and $m$ are coprime.

Can someone help me with this?

user26857
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user10024395
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2 Answers2

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Hint: $a,m$ coprime implies the existence of integers $r,s$ s.t. $ra+sm=1$ (lemma of Bezout).

Hint2:

If $ra=1$ mod $m$ then $m|ra-1$ so that $ra-1=sm$ for some integer $m$

Vera
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  • i can prove this one. My problem is the other way round, which is if multiplicative inverse exists, then a,m are cop rime – user10024395 Aug 14 '14 at 14:33
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    The existence of multiplicative inverse implies the existence of integer $r,s$ with $ra+sm=1$. If $d=gcd(a,m)$ then $d$ divides $ra+sm=1$ so $d$ must be $1$. – Vera Aug 14 '14 at 14:38
  • @Vera "for some integer $m$" should be "for some integer $s$." – rndm_me May 15 '24 at 09:05
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Suppose there exists a multiplicative inverse q such that

$aq \equiv 1\ (mod\ m) \implies aq - 1 = mk \implies aq + mk_1 = 1 \implies gcd(a,m) = 1$ can be easily proven I will leave it as excerise to you how I reached that last step.

Now going the other side

Suppose that $gcd(a,m) = 1 \implies ay_1 + mq_1 = 1 \implies ay_1 \equiv 1 \ (mod\ m)$, so multiplicative inverse exists.

I have taken few short-cuts since some steps are trivial to complete but if you don't see it right away I can modify my answer.

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    would you mind explicitly stating how you got to the gcd(a, m)=1 step in your first proof? – notReallyDev Aug 05 '17 at 19:42
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    You can do it directly. Constructing gcd is as follows you can easily prove that $gcd(a,b) = min{ au + mv : au + mv > 0}$ Here we are considering the integers by well ordering principle such number exist. As integers begin with 1, so that is minimum such possible. –  Aug 07 '17 at 19:22
  • You can do a more elementary argument, but this is as deep as you can get. –  Aug 07 '17 at 19:22