Modulus in Congruence

Question

I was proving a function to be onto but got stuck at point

(7 mod 15) * (? mod 15) = 1 mod 15 I need some value at ? .So that I can get 1 mod 15.

Thank you very much . Every help is appreciated

Answer 1

In order for the modular inverse of $7\bmod 15$ to exist at all, you need $\gcd(7,15)=1$, which is true here.

Since $\color{red}{2}\cdot 7\equiv 14\equiv \color{red}{-1} \bmod 15$, you will have $\color{red}{-2}\cdot 7\equiv 1 \bmod 15$. And $-2\equiv 13\bmod 15$.