Use the methods of contrapositive proof to prove the followi

Use the methods of contrapositive proof to prove the following statements. Let a elementof Z, n elementof N. If a has remainder r when divided by n, then a identicalto r (mod n).

Solution

Let, a not be r mod n

So,

a=r\' mod n where r is not equal to r\'

So, a-r\'=0 mod n

Hence, a-r\' is a multiple of n

a-r\'=kn for some integer k

So, a=kn+r\'

Hence, a give remainder r\' on being divided by a

Hence,a does not give remainder r on being divided by n

Hence proved