PLEASE SHOW DETAIL WILL RATE LIFESAVER if done so!! Let a, b. and c be integers, let n be a natural number, and suppose c 0. If ac. = bc (mod n). then a = b (mod n). Suggestion: We showed in class that this is false by finding a counterexample. Make sure you understand the counterexample. Then, for n = 6, work out choices for a, b, and c. that give another counterexample. Solution Counter example : n = 6 , c = 4 , a = 4 , b = 1 16 = 4 (mod 6) true, but 4 = 1(mod 6) false So, given statement is false. Given statement is true if gcd(c,n) = 1 Proof : Given, ac = bc (mod n) So, ac = nk + bc c(a - b) = nk so nk is a multiple of c. If gcd(c,n) = 1 , then k will be a multiple of c So, let k = cl ac = ncl + bc a = nl + b a = b (mod n).