The power of 2... again.

Algebra Level 2

( 2 + 1 ) ( 2 2 + 1 ) ( 2 4 + 1 ) ( 2 8 + 1 ) ( 2 16 + 1 ) ( 2 32 + 1 ) + 1 = 2 n (2 + 1)(2^2 + 1)(2^4 + 1)(2^8 + 1)(2^{16} + 1)(2^{32} + 1) + 1 = 2^n

Find n n .

This is part of the series: " It's easy, believe me! "


The answer is 64.

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Thành Đạt Lê by Thành Đạt Lê , 17, Singapore

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1 solution

Chew-Seong Cheong
Aug 15, 2017

X = ( 2 + 1 ) ( 2 2 + 1 ) ( 2 4 + 1 ) ( 2 8 + 1 ) ( 2 16 + 1 ) ( 2 32 + 1 ) + 1 = ( 2 2 − 1 ) ( 2 2 + 1 ) ( 2 4 + 1 ) ( 2 8 + 1 ) ( 2 16 + 1 ) ( 2 32 + 1 ) + 1 = ( 2 4 − 1 ) ( 2 4 + 1 ) ( 2 8 + 1 ) ( 2 16 + 1 ) ( 2 32 + 1 ) + 1 = ( 2 8 − 1 ) ( 2 8 + 1 ) ( 2 16 + 1 ) ( 2 32 + 1 ) + 1 = ( 2 16 − 1 ) ( 2 16 + 1 ) ( 2 32 + 1 ) + 1 = ( 2 32 − 1 ) ( 2 32 + 1 ) + 1 = 2 64 − 1 + 1 = 2 64 \begin{aligned} X & = (2+1)(2^2+1)(2^4+1)(2^8+1)(2^{16}+1)(2^{32}+1)+1 \\ & = (2^2-1)(2^2+1)(2^4+1)(2^8+1)(2^{16}+1)(2^{32}+1)+1 \\ & = (2^4-1)(2^4+1)(2^8+1)(2^{16}+1)(2^{32}+1)+1 \\ & = (2^8-1)(2^8+1)(2^{16}+1)(2^{32}+1)+1 \\ & = (2^{16}-1)(2^{16}+1)(2^{32}+1)+1 \\ & = (2^{32}-1)(2^{32}+1)+1 \\ & = 2^{64}-1+1 \\ & = 2^{64} \end{aligned}

⟹ n = 64 \implies n = \boxed{64}

11 Helpful 0 Interesting 0 Brilliant 0 Confused

Beautifully solved...

James Bacon - 2 years, 10 months ago

wish I had seen that, nice one…

Dominik A. Schäfer - 2 years, 10 months ago

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