Use Euclid's Algorithm to find the multiplicative inverse Use Euclid's Algorithm to find the multiplicative inverse of $13$ in $\mathbf{Z}_{35}$
Can someone talk me through the steps how to do this? I am really lost on this one. 
Thanks 

Basically you want to find $a$ such that $13a \equiv 1 \mod 35$ which is the same as: $$13a + 35k = 1, \qquad \text{For some }k\in \mathbb Z$$
Use Euclid's Algorithm on $13$ and $35$, the same way as for finding $\gcd(13,35)$.
So start with $35 = 2\cdot13 + 9$ and so on... Then substitute your answers in the line above (called reversing Euclid's Algorithm) and you'll find $a$ and $k$.
Your answer is $a$.
