Question

N is a positive integer. Show that, for all n, (7n + 3)² - (7n - 3)² is a multiple of 84. ​

Answer 1

Use the difference of squares rule.

a^2 - b^2 = (a-b)(a+b)

(7n+3)^2 - (7n-3)^2 = ( (7n+3)-(7n-3) )( (7n+3)+(7n-3) )

(7n+3)^2 - (7n-3)^2 = (7n+3-7n+3)(7n+3+7n-3)

(7n+3)^2 - (7n-3)^2 = (6)(14n)

(7n+3)^2 - (7n-3)^2 = (6*14)n

(7n+3)^2 - (7n-3)^2 = 84n

This proves (7n+3)^2 - (7n-3)^2 is a multiple of 84 because 84 is a factor of 84n.