There are two nice methods.
The first is by applying the Bezout theorem. This states that if ##a## and ##b## are integers, then there are integers ##p## and ##q## such that ##pa+bq = \text{gcd}(a,b)##. In this case we take ##a=108## and ##b=625##. The gcd is ##1## (which is a necessary and sufficient condition for finding inverses). So we search for integers ##p## and ##q## such that ##108p + 625q=1##. Our ##p## will be the inverse (just look at the equation (mod 625)).
How to find ##p## and ##q##? Well, we have the Euclidean algorithm for finding the gcd. So let's do that:
##625 = 108\cdot 5 + 85##
##108 = 85\cdot 1 + 23##
##85 = 23\cdot 3 + 16##
##23 = 16\cdot 1 + 7##
##16 = 7\cdot 2 + 2##
##7 = 2\cdot 3 + 1##
See
http://en.wikipedia.org/wiki/Euclidean_algorithm for the details if this is unfamiliar. So we see indeed that the gcd is ##1##. Now let us do the Euclidean algorithm in reverse:
<br />
\begin{eqnarray*}<br />
1 & = & 7 - 2\cdot 3\\<br />
& = & 7 - (16 - 7\cdot 2)\cdot 3\\<br />
& = & 7\cdot (1 + 2\cdot 3) - 16\cdot 3\\<br />
& = & 7\cdot 7 - 16\cdot 3\\<br />
& = & (23 - 16\cdot 1)\cdot 7 - 16\cdot 3\\<br />
& = & 23\cdot 7 - 16\cdot (1\cdot 7 + 3)\\<br />
& = & 23\cdot 7 - 16\cdot 10\\<br />
& = & 23\cdot 7 - (85 - 23\cdot 3)\cdot 10\\<br />
& = & 23\cdot (7 + 3\cdot 10) - 85\cdot 10\\<br />
& = & 23\cdot 37 - 85\cdot 10\\<br />
& = & (108 - 85\cdot 1)\cdot 37 - 85\cdot 10\\<br />
& = & 108\cdot 37 - 85\cdot (1\cdot 37 +10)\\<br />
& = & 108\cdot 37 - 85\cdot 47\\<br />
& = & 108\cdot 37 - (625 - 108\cdot 5)\cdot 47\\<br />
& = & 108\cdot (37 + 5\cdot 47) - 625\cdot 47\\<br />
& = & 108\cdot 272 - 625\cdot 47<br />
\end{eqnarray*}
Thus we get
1 = 108\cdot 272 - 625\cdot 47
Looking at this equation in (mod 625), we get ##108\cdot 272 = 1##. So the inverse of 108 is 272.
A second method is by Euler's theorem (the generalization of Fermat's little theorem): This states that if ##\text{gcd}(a,n)=1##, then ##a^{\varphi(n)} = 1## in (mod n). In our case, we get ##108^{\varphi(625)} = 1## in (mod 625). So the inverse is ##108^{\varphi(625)-1}##. Now we need to calculate the Euler totient function ##varphi##. There are several methods, but let us use this one:
http://en.wikipedia.org/wiki/Euler's_totient_function#Euler.27s_product_formula
So since ##625 = 5^4##, we get
\varphi(625) = 625\cdot (1 - \frac{1}{5}) = 625 - 125 = 500
So the inverse is ##108^{499}##. This is an easier method to calculate the inverse, but it is more annoying since it leaves us with a tough exponential to calculate. A somewhat similar method but with smaller exponential can be found here
http://en.wikipedia.org/wiki/Carmichael_function