WMC 2018 Senior Problem 30

Algebra Level 3

Which of the following is not a divisor of 2 18 − 1 2^{18}-1 ?

2 9 + 1 2^{9}+1 2 6 − 1 2^{6}-1 2 3 + 1 2^{3}+1 They are all divisors 2 3 − 1 2^{3}-1

Tiger Ang by Tiger Ang , 20, Malaysia

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1 solution

Julian Yu
Oct 20, 2018

2 18 − 1 = ( 2 9 + 1 ) ( 2 9 − 1 ) 2^{18}-1=(2^9+1)(2^9-1) .

Also, 2 18 − 1 = ( 2 6 − 1 ) ( 2 12 + 2 6 + 1 ) = ( 2 3 + 1 ) ( 2 3 − 1 ) ( 2 12 + 2 6 + 1 ) 2^{18}-1=(2^6-1)(2^{12}+2^6+1)=(2^3+1)(2^3-1)(2^{12}+2^6+1) . Therefore, all of the choices are factors.

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