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# What is the greatest prime factor of 3^6 - 1 ?

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Manager
Joined: 30 May 2013
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What is the greatest prime factor of 3^6 - 1 ?  [#permalink]

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Updated on: 04 Feb 2014, 02:35
1
13
00:00

Difficulty:

35% (medium)

Question Stats:

71% (01:22) correct 29% (01:27) wrong based on 779 sessions

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What is the greatest prime factor of 3^6 - 1 ?

A. 2
B. 3
C. 7
D. 13
E. 17

Originally posted by rrsnathan on 01 Sep 2013, 17:59.
Last edited by Bunuel on 04 Feb 2014, 02:35, edited 1 time in total.
Edited the question.
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Re: What is the greatest prime factor of the number (3^6 - 1) ?  [#permalink]

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01 Sep 2013, 21:00
15
12
rrsnathan wrote:
What is the greatest prime factor of the number (3^6 - 1) ?
a) 2
b) 3
c) 7
d) 13
e) 17

Apply $$a^2-b^2=(a-b)(a+b)$$.

$$3^6 - 1=(3^3-1)(3^3+1)=26*28=(2*13)*(2^2*7)$$ --> the greatest prime factor is 13.

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Re: What is the greatest prime factor of the number (3^6 - 1) ?  [#permalink]

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01 Sep 2013, 20:44
1
3^6-1 = (3^3)^2 -1 = (27^2)-1

Dividing (27^2)-1 by 13 will give us a reminder of 0 ( Hint: (2*13+1)^2-1/13 = (Reminder 1)-1=0

Hence the greatest prime factor must be 13.

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Re: What is the greatest prime factor of 3^6 - 1 ?  [#permalink]

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23 Jan 2015, 13:47
4
Hi All,

Both Smallwonder and Bunuel have provided elegant solutions to this question. The basic math behind this question is Arithmetic and Prime Factorization though, so if you don't immediately "see" the elegant approach, you can still get to the answer....

We're asked to find the LARGEST prime factor of 3^6 - 1

3^6 = 9^3 = (9)(9)(9) = 729

729 - 1 = 728

Now, we can prime factor 728.

You probably immediately see that 728 is divisible by 2, but if you know your 'rules of division', you can see that it's also divisible by 4...

728 =
(4)(182)
(4)(2)(91)
(4)(2)(7)(13)

13 is the largest prime.

Sometimes this type of approach isn't practical (especially if the numbers involved are HUGE), but when you're given 'manageable' numbers, there's nothing wrong with admitting that you don't see the 'hidden pattern.' If you can get to the correct answer in a reasonable amount of time by just doing arithmetic, then it's better to do THAT than waste time staring at the screen.

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Re: What is the greatest prime factor of 3^6 - 1 ?  [#permalink]

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30 Jan 2015, 00:18
1
$$3^6 - 1$$

$$= 9^3 - 1^3$$

$$= (9-1)(9^2 + 9 + 1)$$

$$= 8 * 91$$

$$= 8 * 13 * 7$$

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Re: What is the greatest prime factor of 3^6 - 1 ?  [#permalink]

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30 Jan 2015, 04:20
hi i think a simpler and easier method could be..
3^6-1=3^6-1^6 = (3^3-1)(3^3+1)
=26*28... clearly 13 is the ans
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3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html
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Re: What is the greatest prime factor of 3^6 - 1 ?  [#permalink]

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16 Mar 2016, 01:33
Using the identity A^2 - B^2 = A+B x A-B
we get 28 x 26 => 13 hence D
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Re: What is the greatest prime factor of 3^6 - 1 ?  [#permalink]

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16 Mar 2016, 04:12
rrsnathan wrote:
What is the greatest prime factor of 3^6 - 1 ?

A. 2
B. 3
C. 7
D. 13
E. 17

It seems it does not have any pattern. The greatest prime factor changes randomly as the power increases.
So, I found the value of the expression and prime factored. That gives 13.
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Re: What is the greatest prime factor of 3^6 - 1 ?  [#permalink]

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20 Apr 2016, 12:50
Mathivanan Palraj wrote:
rrsnathan wrote:
What is the greatest prime factor of 3^6 - 1 ?

A. 2
B. 3
C. 7
D. 13
E. 17

It seems it does not have any pattern. The greatest prime factor changes randomly as the power increases.
So, I found the value of the expression and prime factored. That gives 13.

Using the identity will be way easier here ..!!
regards
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Re: What is the greatest prime factor of 3^6 - 1 ?  [#permalink]

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04 May 2016, 12:22
$$3^{6}$$ - $$1$$ = $$729$$ - $$1$$
= $$728$$
prime factors of $$728$$ =$$2^{3}$$ * $$7$$ * $$13$$
greatest prime factor is 13
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Re: What is the greatest prime factor of 3^6 - 1 ?  [#permalink]

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11 Dec 2017, 07:39
Top Contributor
rrsnathan wrote:
What is the greatest prime factor of 3^6 - 1 ?

A. 2
B. 3
C. 7
D. 13
E. 17

3^6 - 1 is a difference of squares, since 3^6 = (3³)² and 1 = 1²

So, we get: 3^6 - 1 = (3³ + 1)(3³ - 1)
= (27 + 1)(27 - 1)
= (28)(26)
= (2)(2)(7)(2)(13)

The prime factors are 2, 7 and 13
The greatest value is 13

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Re: What is the greatest prime factor of 3^6 - 1 ?  [#permalink]

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28 Jan 2019, 08:25
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Re: What is the greatest prime factor of 3^6 - 1 ?   [#permalink] 28 Jan 2019, 08:25
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