Prove that the relation of congruence modulo n is transitive. That is, prove that for all integers a, b, c, and n with n> 1, if a =b (mod n) and bc (mod n), then aAC (mod n). Proof: Suppose a, b, c, andn are any integers with n> 1 and a =b (mod n) and b c(mod n). By definition of congruence modulo n, this means that n| (a - b) and n By definition of divisibility, since n| (a - b), and since n there are integers r and s such that a - b= m and b - sn. Now a -c = (a- b)+ Rewriting the difference on the right-hand side of this equation in terms of n, r, and s and factoring the result completely gives that a-c= definition of divisibility. Thus a = c (mod n) by definition of congruence modulo n. Since the sum of any two integers is an integer, it follows that n

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