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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≡1and ∃x,y∈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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Most popular questions from this chapter

A d-ary tree is a rooted tree in which each node has at most dchildren. Show that any d-ary tree with n nodes must have a depth ofΩ(lognlogd) .Can you give a precise formula for the minimum depth it could possibly have?

The algorithm for computing abmodCby repeated squaring does not necessarily lead to the minimum number of multiplications. Give an example of b>10where the exponentiation can be performed using fewer multiplications, by some other method.

1.36. Square roots. In this problem, we'll see that it is easy to compute square roots modulo a prime pwith p≡3(mod4).
(a) Suppose p≡3(mod4). Show that(p+1)/4 is an integer.
(b) We say x is a square root of a modulo p if a=x2(modp). Show that if p≡3(mod4)and if a has a square root modulo p, thena(p+1)/4 is such a square root.

Prove that the grade-school multiplication algorithm (page 24), when applied to binary numbers, always gives the right answer.

Give an efficient algorithm to compute the least common multiple of two n-bit numbers x and y, that is, the smallest number divisible by bothx and y. What is the running time of your algorithm as a function of n?
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