Posted by **Michael** on Thursday, February 14, 2013 at 3:48am.

Prove that if ab = ac (mod n) and a is relatively prime to n, then b = c (mod n).

Proof: a and n are relatively prime and from ab = ac(mod n), we have n|(ab-ac), so n|a(b-c). Since (a,n)=1 (relatively prime), we get n(b-c). hence b=c(mod n).

But what if a and n are not relatively prime, can you still prove ab = ac (mod n)? Can you show a counterexample if I cannot be done? Thank you.

- Math (Proof) -
**Steve**, Thursday, February 14, 2013 at 4:24am
if not relatively prime, no proof.

2*3 (mod 8) = 2*7 (mod 8)

but not 3 = 7 (mod 8)

The primeness is vital. n can divide ab-ac of the products, but if a factor of n is also factor of a, then n need not divide b-c.

- Math (Proof) -
**Michael**, Thursday, February 14, 2013 at 5:32am
Thank you!

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