Question

prove that one and only one out of n, n+2 and n, +4 is divisioble by 3,where n is any positive integer ​

Answer 1

SOLUTION:Any positive integer is of the form 3q or, 3q+1 or, 3q+2 for some integer q and one and only one of these possibilities can occur.

CASE 1 when n=3q : In this case,we have

n=3q, which is divisible by 3

Now, n=3q

=n+2=3q+2,

=n+2 leaves remainder 2 when divided by 3.

=n+2 is not divisible by 3.

CASE 2 when n=3q+1 : In this case,we have

n=3q + 1

=n leaves remainder 1 when divided by 3.

=n is not divisible by 3

CASE 3 when n=3q+2:in this case we have

n=3q+2

=n+2= 3q+2+2=3(q+1)+1

=n+2 leaves remainder 1 when divided by 3

=n+2 is not divisible by 3.

Thus, n, n+2, n+4 are not divisible by 3.

Hope this is helpful to you ..