Problems need HardWork #6

Number Theory Level 3

Evaluate gcd ⁡ ( 2 120 − 1 , 2 100 − 1 ) \gcd(2^{120} - 1, 2^{100} - 1) .

1 1 2 2 2 20 − 1 2^{20} - 1 2 100 − 1 2^{100} - 1

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Dev Sharma by Dev Sharma , 20, India

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1 solution

Vishnu Bhagyanath
Aug 22, 2015

Application of the formula that g c d ( a m − 1 , a n − 1 ) = a g c d ( m , n ) − 1 gcd(a^m -1, a^n -1) = a^{gcd(m,n)}-1 , proof of which can be found here .

g c d ( 2 120 − 1 , 2 100 − 1 ) = 2 g c d ( 120 , 100 ) − 1 = 2 20 − 1 gcd(2^{120} -1,2^{100} -1) = 2^{gcd(120,100)}-1 = 2^{20}-1

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