Prove by either contradiction or contraposition a Z if a mod

Prove by either contradiction or contraposition: a Z, if a mod 6 = 3 then a mod 3 = 2.

Solution

We prove by contraditionc

Assume a mod 6=3 and a mod 3=2

So, a=3k+2 , for some integer k

But, a mod 6=3

So, 3k+2=3 mod 6

3k=1 mod 6

A multiple of 3 can given 0 or 3 as remainder modulo 6

So , 3k=1 is not possible

Hence a contradictio

So, a mod 3 is not equal to 2