Question

Prove that 16^4–2^13–4^5 is divisible by 11

Answer 1

Explanation:

(16^4 -2^13 -4^5) mod 11 = ((2^4)^4 -2^13 -(2^2)^5) mod 11

= (2^16 -2^13 -2^10) mod 11

= (2^10 mod 11)·((2^6 -2^3 -1) mod 11)

= (1024 mod 11)·(55 mod 11)

= 1 · 0 = 0

When the sum shown above is divided by 11, its remainder is zero. Hence the sum is divisible by 11.

Answer 2

$16^4-2^{13}-4^5=2^{16}-2^{13}-2^{10}=2^{10}(2^6-2^3-1)=2^{10}(64-8-1)=\\=2^{10}\cdot55=2^{10}\cdot5\cdot11$