What is the bank need to get the message?

Question

In Number theory $p=37, q= 43$, $\phi(pq)= 36 \cdot 42$, $e=5$ $d=?$

What does the bank need to get the message? I don't understand this problem. Can any one help me please?

Answer 1

This is a simple RSA problem: $d$ is the inverse of $e$ modulo $\phi(n)$, where $n = pq$

So apply the extended Euclidean algorithm (see here) to $e$ and $\phi(n)$.

You get integers $k,l \in \mathbb{Z}$ with $ke + l\phi(n) = 1$.

Then $k$ is easily seen to be the required inverse, i.e. $d$ (just take the previous equation modulo $\phi(n)$ where the right term vanishes).