3. (Linear Congruence). An equation in integers of the form ax=b (mod m) is called a linear congruence. We want to find an integer x that will satisfy it. (Note that ax=b (mod m) is equivalent to saying ax-b=nm for some integer n.
a) Prove that if (a, m) = 1 then there is always a solution. (Hint: express the gcd as a linear combination and multiply by b.).
b) For 56 and 15 use the Euclidean algorithm to express the gcd 1= (56, 15) as a linear combination 56s+15t=1.
c) Use part b to find a value of x so that 15x=7 (mod 56).
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The proof has two parts: 1) existence of a solution and 2) uniqueness of the solution.
For the existence part, we must effectively build the solution.
Since (a,m)=1 => exists integers p and q such that a*p+m*q=1
We multiply the previous relation with b in both sides => b*a*p +b*m*q= b
Reducing the last equation modulo m => b*a*p = b (mod m) which is equivalent with a(bp)= b (mod m)
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