What does it mean to “have a multiplicative inverse of modulo 10!”?

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

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?
• @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 $\rm\,> n.\:$