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Here's the question:

What's the smallest integer > 1 that has a multiplicative inverse modulo 10! (that is, 10 factorial)?

What does that mean?

I understand that:

We say that x is the multiplicative inverse of a modulo N whenever ax = 1(mod N)

But... we only know N in this case right? I'm very confused

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up vote 3 down vote accepted

You have the right definition. And it is a theorem that $a$ has a multiplicative inverse modulo $m$ if and only if $a$ and $m$ are relatively prime. So what is the smallest integer $a\gt 1$ such that $a$ and $10!$ are relatively prime?

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So, if I were to say A has a multiplicative inverse modulo B, this means that there exists an X such that Ax(mod B) = 1 right? – Jess Mar 21 '13 at 2:41
@Jess: Exactly. But in order to answer the specific question, you need some information about which numbers have a multiplicative inverse modulo $m$. These are the $a$'s relatively prime to $m$. In the case of $m=10!$, the first number after $1$ that has an inverse modulo $m$ is $11$, the next is $13$, then $17$, then $19$, then $23$, and so on. They are definitely not all prime, but the first non-prime one is $11^2$. – André Nicolas Mar 21 '13 at 2:46

Hint $\ $ By Bezout, $\rm\ a\ $ is invertible mod $\rm\:n!\:$ iff $\rm\ a\ $ is coprime to $\rm\:n!.\:$ Thus if $\rm\ a>1\:$ then every prime divisor of $\rm\ a\ $ is $\rm\, > n,\:$ e.g. the next prime $\rm\,> n.\:$

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