Question

Prove the divisibility of each of the following numbers: 7^6 + 7^5 - 7^4 by 11

Answer 1

7^4 * 5

You need to factor the answer.

Answer 2

`7^6 + 7^5 - 7^4\\\\7^4*7^2 + 7^4*7 - 7^4*1\\\\7^4(7^2 + 7 - 1)\\\\7^4(49 + 7 - 1)\\\\7^4(55)\\\\7^4(11*5)\\\\11*5*7^4\\\\11k\ \text{ where } k = 5*7^4\\\\`

The result we end up with is 11k which is a multiple of 11.

Therefore, this proves that
`7^6 + 7^5 - 7^4` is divisible by 11.