Prove that for all integers a, b, and c. If there exist integers m and n such that am + bn =1 and c cannot be equal to 1 or negative 1, then c does not divide a or c does not divide b.
This is the problem 2(d) in the section 1.6, page 60 of the book "A Transition to Advanced Mathematics, written by Smith/Eggen/St. Andre. 2014 Edition."
I would like that you give me your opinion about my proof. Thanks!
Proof by Contradiction. Suppose that c divides a, and c divides b. Then, a=ck and b=cj, for some integers k and j. Therefore, am+bn=1=ck+cj=c(k+j). Thus, k+j = 1/c. But this is a contradiction because k, j and c are integers and c cannot be equal to 1 or negative 1. Q.E.D.
Best Answer
am+bn=1=ck+cj=c(k+j) should be $$ am+bn=1=ckm+cjn=c(km+jn)=1$$
And your proof is valid.