Question

Prove or disprove: If a has an inverse modulo b, then b has an inverse modulo a.

Short answer

Yes, It can be proved that ifa has an inverse modulo b, then has an inverse modulo a.

Step-by-step solution

Explain inverse modulo

Modulo is the remainder of the division , that in inverse is added with quotient is called inverse modulo.

Prove the given statement

Consider that if $a$has an inverse modulo $b$Then $\gcd \left(a,b\right)\equiv 1$and $\exists x,y\in \cancel{c}$,such that $ax+by=1$.

Thus $x$ is one of the inverse elements of $a$modulo b.

By symmetry, y is one of the inverse elements of b modulo a.

Therefore, it is proved that if a has an inverse modulo b, then b has an inverse modulo a.