Question

Prove the divisibility of the following number using factoring!
7^6 +7^5 -7^4 by 11

Answer 1

x = 7^6 + 7^5 - 7^4
x = 7^4*7^2 + 7^4*7 - 7^4*1
x = 7^4*(7^2 + 7 - 1)
x = 7^4*(49 + 7 - 1)
x = 7^4*55
x = 55*7^4
x = 11*5*7^4
x = 11*y .... where y = 5*7^4 is an integer

Since x = 11*y, and y is an integer, this means x is a multiple of 11

So that proves 7^6 + 7^5 - 7^4 is divisible by 11