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

Question

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

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Answer 1 · 2018-04-02T11:53:31

See below...

#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#.