Question

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

Answer 1

`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`

Answer 2

Step-by-step 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.