   Chapter 2.5, Problem 6E

Chapter
Section
Textbook Problem

# Find a solution x ∈ ℤ , 0 ≤ x < n , for each of the congruences a x ≡ b   ​ ( mod   n ) in Exercises 3 − 24 .Note that in each case, a and n are relatively prime. 3 x ≡ 4   ( mod   13 )

To determine

A solution x,0x<13, for the congruence 3x4(mod13) where 3 and 13 are relatively prime.

Explanation

Given Information:

3 and 13 are relatively prime.

Formula used:

1) Theorem: If a and n are relatively prime, the congruence axb(modn) has a solution x in the integers, and any two solutions in are congruent modulo n.

2) The Euclidean Algorithm:

a=bq0+r1,0r1<bb=r1q1+r2,0r2<r1r1=r2q2+r3,0r3<r2rk=rk+1qk+1+rk+2,0rk+2<rk+1

Since the integers r1,r2,,rk+2 are decreasing and are all non-negative, there is a smallest integer n such that rn+1=0: rn1=rnqn+rn+1,0=rn+1.

Explanation:

When a and n are relatively prime, the Euclidean Algorithm can be used to find a solution x to

axb(modn)

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