What is the the value of a? Also, prove the following:

Prove that 2^36-1 is divisible by 9 and if 2^36-1=68a19476735 then find the value of a.

Thank you!!

1 Answer
Apr 2, 2018

See below...

Explanation:

2^36-1

Apply difference of square rule,

(2^18-1)(2^18+1)

Further apply difference of square rule,

(2^9-1)color(blue)((2^9+1))(2^18+1)

Apply difference of cube rule,

(2^9-1)color(blue)((2^3+1)(2^6-2^3+1))(2^18+1)

Notice that (2^3+1) is a factor of 2^36-1,

2^3+1=9

Hence, 9 is a factor of 2^36-1 and thus, 2^36-1 is divisible by 9.