Question

Prove that for all whole values of n the value of the expression:
n(n+2)–(n–7)(n–5) is divisible by 7.

Answer 1

Explanation:

Multiply it out, collect terms, and look for a factor of 7:

  n(n +2) -(n -7)(n -5) = n² +2n -(n² -12n +35)

  = 14n -35

  = 7(2n -5)

The expression has a factor of 7, so is divisible by 7 with a resulting quotient of 2n-5.