Question

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

it needs to bein exponential form

Answer 1

the core is 7⁴.

so, we have

7⁴×7² + 7⁴×7 - 7⁴

7⁴ = 3 mod 11 (remainder of 3 when dividing by 11)

7² = 5 mod 11

7 = 7 mod 11

7⁴×7² = 3×5 mod 11 = 15 mod 11 = 4 mod 11

7⁴×7 = 3×7 mod 11 = 21 mod 11 = 10 mod 11

so,

7⁴×7² + 7⁴×7 - 7⁴ =

(4 + 10 - 3) mod 11 = 11 mod 11 = 0 mod 11

therefore,

7⁶ + 7⁵ - 7⁴ is divisible by 11