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# If n is a positive integer, greater than 2, what is the greatest prime

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If n is a positive integer, greater than 2, what is the greatest prime  [#permalink]

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25 Nov 2018, 05:33
1
4
00:00

Difficulty:

65% (hard)

Question Stats:

57% (01:30) correct 43% (01:58) wrong based on 37 sessions

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If n is a positive integer, greater than 2, what is the greatest prime factor of $$3^n+3^n+3^n-3^n^-^2?$$

A. 3
B. 5
C. 7
D. 11
E. 13
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Re: If n is a positive integer, greater than 2, what is the greatest prime  [#permalink]

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25 Nov 2018, 06:02
1
$$3^n+3^n+3^n-3^{n-2}$$

= $$3^n (1+1+1-\frac{1}{3^2})$$

= $$3^n (3-\frac{1}{9})$$

= $$3^n (26/9)$$

Hence the largest prime factor is 13.

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Re: If n is a positive integer, greater than 2, what is the greatest prime  [#permalink]

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25 Nov 2018, 12:09
rencsee wrote:
If n is a positive integer, greater than 2, what is the greatest prime factor of $$3^n+3^n+3^n-3^n^-^2?$$

A. 3
B. 5
C. 7
D. 11
E. 13

$$?\,\,\,:\,\,\,\,3 \cdot {3^n} - {3^{n - 2}} = {3^{n + 1}} - {3^{n - 2}}\,\,\,{\rm{max}}\,\,{\rm{prime}}\,\,{\rm{factor}}\,\,\,\,\,\,\,\left[ {n \ge 3\,\,{\mathop{\rm int}} \,\,\left( * \right)} \right]$$

$${3^{n + 1}} - {3^{n - 2}} = {3^{n - 2}}\left( {{3^3} - 1} \right) = {3^{n - 2}} \cdot 2 \cdot 13\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\,? = 13\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left( {\rm{E}} \right)\,\,\,\,\,\,$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: If n is a positive integer, greater than 2, what is the greatest prime &nbs [#permalink] 25 Nov 2018, 12:09
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