Thursday, 5 February 2015

modular arithmetic - 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


Answer



Hint: $13$ and $35$ are relatively prime. Use the extended Euclidean algorithm to find the integers $x$ and $y$ such that:



$$13x + 35y = 1$$



From here, simply mod out by $35$:




$$13x \equiv 1 \pmod{35}$$


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