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Difference between revisions of "1971 AHSME Problems/Problem 14"

(Created page with "== Problem == The number <math>(2^{48}-1)</math> is exactly divisible by two numbers between <math>60</math> and <math>70</math>. These numbers are <math>\textbf{(A) }61,63\...")
(No difference)

Revision as of 18:23, 28 January 2021

Problem

The number $(2^{48}-1)$ is exactly divisible by two numbers between $60$ and $70$. These numbers are

$\textbf{(A) }61,63\qquad \textbf{(B) }61,65\qquad \textbf{(C) }63,65\qquad \textbf{(D) }63,67\qquad \textbf{(E) }67,69$

Solution

Factor. \[2^{48}-1 = (2^{24}+1)(2^{12}+1)(2^{6}+1)(2^{3}+1)(2^{3}-1)\]

We only care about two terms: $2^{6}+1$ and $(2^{3}+1)(2^{3}-1)$. These simplify to $65$ and $63.$

The answer is $\textbf{(C)}.$

-edited by coolmath34

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