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Chapter 1: Q22E (page 49)

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

Short Answer

Expert verified

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

Step by step solution

01

Explain inverse modulo

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

02

Prove the given statement

Consider that if ahas an inverse modulo bThen gcda,b鈮�1and 鈭�x,y鈭�c,such that ax+by=1.

Thus x is one of the inverse elements of amodulo 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.

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