"multiplicative inverse in the modulo of the larger number" what does that mean? while I was reading this artical I have read the following paragraph: 


  The interesting thing is that if two numbers have a $\gcd$ of $1$, then the smaller of the two numbers has a multiplicative inverse in the modulo of the larger number. It is expressed in the following equation:


and he gives the following example:

Lets work in the set $\mathbb{Z_9}$, then $4\in\mathbb{Z_9}$ and $\gcd(4,9)=1$.

Therefore $4$ has a multiplicative inverse (written $4^{−1}$) in $\bmod9$, which is $7$.

And indeed, $4\cdot7=28\equiv1\pmod9$.

But not all numbers have inverses.

For instance, $3\in\mathbb{Z_9}$ but $3^{−1}$ does not exist!

This is because $\gcd(3,9)=3\neq1$.

but what I do not understand is what does he mean by:


  then the smaller of the two numbers has a multiplicative inverse in
  the modulo of the larger number.


and how he got the $7$
The two numbers in his example are $4$ and $9$. The statement is that $4$ has a multiplicative inverse in the integers modulo $9$, or in other words, there is an integer $n$ such that $4 \cdot n \equiv 1 \mod 9$. The $7$ can be obtained by some trial and error (you only need to check the integers $1$ through $9$). He then gives an example of an integer that does not have a multiplicative inverse modulo $9$, namely $3$.
A multiplicative inverse of a number $a\in R$ for a ring $R$ is a number $a^{-1}$ such that $aa^{-1}=1$.

What he means by saying "modulo the bigger number" which we can denote $b$ is a multiplicative inverse of $a$ (the smaller number) in the ring $\mathbb{Z_b}$ which is the ring of the integers under the identification $\forall c,d \in \mathbb{Z} \ \ \bar{c}=\bar{d}\in\mathbb{Z}_b \  \iff c\equiv d(\mbox{mod} b) $. 

Note this is intereseting because in general $\mathbb{Z}_b$ is a ring where we do not know if we have multiplicative inverses.

$\mathbb{Z}_b$ is a field (commutative ring with inverses) $\iff \ b$ is a prime number.