Question

product of three consecutive integer is divisible by 6 : Is it true ? Why ?​

Answer 1

Let n, n+1, n+2 be the three consecutive positive integer. We know that any positive integer is of the 6q,6q+1, 6q+2, 6q+3, 6q+4 or 6q+5.

If n=6q, then

n(n+1)(n+2)=6q(6q+1)(6q+2), which is divisible by 6.

If n=6q+1, then

n(n+1)(n+2)=(6q+1)(6q+2)(6q+3)

{Taking common outside}

n(n+1)(n+2)=6(6q+1)(3q+2)(2q+1), which is divisible by 6.

If n=6q+2, then

n(n+1)(n+2)=(6q+2)(6q+3)(6q+4)

=12(3q+1)(2q+1)(3q+2), which is divisible by 6.

If n=6q+3, then

n(n+1)(n+2)=(6q+3)(6q+4)(6q+5)

=6(2q+1)(3q+2)(6q+5), which is divisible by 6.

If n=6q+4, then

n(n+1)(n+2)=(6q+4)(6q+5)(6q+6)

=12(3q+2)(6q+5)(q+1), which is again divisible by 6.

If n=6q+5, then

n(n+1)(n+2)=(6q+5)(6q+6)(6q+7)

=6(6q+5)(q+1)(6q+7) which is also divisible by 6.