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If 1+2+2^2+... +2^n=2^{n+1}-1, what is the largest prime factor of 1+2

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If 1+2+2^2+... +2^n=2^{n+1}-1, what is the largest prime factor of 1+2  [#permalink]

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New post 27 Nov 2018, 01:40
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A
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C
D
E

Difficulty:

  35% (medium)

Question Stats:

74% (00:44) correct 26% (01:50) wrong based on 19 sessions

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[Math Revolution GMAT math practice question]

If \(1+2+2^2+... +2^n=2^{n+1}-1\), what is the largest prime factor of \(1+2+2^2+... +2^7\)?

\(A. 3\)
\(B. 5\)
\(C. 13\)
\(D. 17\)
\(E. 19\)

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Re: If 1+2+2^2+... +2^n=2^{n+1}-1, what is the largest prime factor of 1+2  [#permalink]

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New post 27 Nov 2018, 01:58
1
\(1+2+2^2+....+2^7\) = 1+ GP series

Sum of GP series = \(\frac{a(r^n-1)}{r-1}\)
a = first term = 2.
r = common ratio = 2.

Sum = \(\frac{2(2^7-1)}{2-1}\) = 254.

Total = 254+1 = 255.
Prime factorizing 255 gives 3*5*17.

Largest prime is 17.

D is the answer.
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Re: If 1+2+2^2+... +2^n=2^{n+1}-1, what is the largest prime factor of 1+2  [#permalink]

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New post 29 Nov 2018, 01:46
1
=>

\(1+2+2^2+... +2^7 = 2^8 – 1 = (2^4+1)(2^4-1) = (2^4+1)(2^2+1)(2^2-1) = (2^4+1)(2^2+1)(2+1)(2-1) = 17*5*3.\)
\(17\) is the largest prime factor of \(1+2+2^2+... +2^7.\)

Therefore, D is the answer.
Answer: D
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Re: If 1+2+2^2+... +2^n=2^{n+1}-1, what is the largest prime factor of 1+2 &nbs [#permalink] 29 Nov 2018, 01:46
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If 1+2+2^2+... +2^n=2^{n+1}-1, what is the largest prime factor of 1+2

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