If possible find a pair of integers x and y that satisfy the

If possible, find a pair of integers x and y that satisfy the given condition. If it is not possible, explain why not.

a) x and   y are congruent modulo 3 but not modulo 6

b) x and y   are congruent modulo 6 but not modulo 3

Solution

  a)
x = 4 and y = 7

4 = 7 (mod 3)

But:
4 ? 7 (mod 6)


b)
Not possible. Assume x = y (mod 6). Then x - y = 6n, where n is some integer. This means x - y = 3*2n, i.e., 3 | x - y, which is true if and only if x = y (mod 3). So if x = y (mod 6) then x = y (mod 3).