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Problem 204

AMC 10/12, early questions

Number theory Difficulty 3.8 Find the answer Lista 6 · Brazil

Problem:

Dois divisores - O número $2^{48} - 1$ é divisível por dois números compreendidos entre 60 e 70. Quais são esses números?
(a) 61 e 63
(b) 61 e 65
(c) 63 e 65
(d) 63 e 67
(e) 67 e 69

Your answer

Official solution

Solution:

Lembre que
$a^{4}-1=(a-1)\left(a^{3}+a^{2}+a+1\right)$
Logo, se $a=2^{12}$ , temos:
$2^{48}-1=\left(2^{12}\right)^{4}-1=\left(2^{12}-1\right)\left(2^{36}+2^{24}+2^{12}+1\right)$
e $2^{12}-1=\left(2^{6}+1\right)\left(2^{6}-1\right)=65 \times 63$ . A opção correta é (c).

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