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

Question

Prove that $2^{36} - 1$ $2^36-1$ is divisible by $9$ $9$ and if $2^{36} - 1 = 68 a 19476735$ $2^36-1=68a19476735$ then find the value of a.

Thank you!!

1351 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-04-02T11:53:31

See below...

$2^{36} - 1$ $2^36-1$

Apply difference of square rule,

$(2^{18} - 1) (2^{18} + 1)$ $(2^18-1)(2^18+1)$

Further apply difference of square rule,

$(2^{9} - 1) (2^{9} + 1) (2^{18} + 1)$ $(2^9-1)color(blue)((2^9+1))(2^18+1)$

Apply difference of cube rule,

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

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

$2^{3} + 1 = 9$ $2^3+1=9$

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