[tex]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\\\\[/tex]
The result we end up with is 11k which is a multiple of 11.
Therefore, this proves that [tex]7^6 + 7^5 - 7^4[/tex] is divisible by 11.
