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

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.

Answer: D.
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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.

Final Answer:

GMAT assassins aren't born, they're made,
Rich

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.

/SW

\(3^6 - 1\)

\(= 9^3 - 1^3\)

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

\(= 8 * 91\)

\(= 8 * 13 * 7\)

Answer = D = 13

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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Using the identity A^2 - B^2 = A+B x A-B
we get 28 x 26 => 13 hence D
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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
Stone Cold
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\(3^{6}\) - \(1\) = \(729\) - \(1\)
= \(728\)
prime factors of \(728\) =\(2^{3}\) * \(7\) * \(13\)
greatest prime factor is 13
Correct answer - D

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

Answer: D

3^6 - 1

Using a² - b²= (a-b)(a+b)


3^6 - 1 = 3^6 - 1^6 = (3³ – 1³) (3³ + 1³)= (27-1)(27+1)=26x28=2x13x2x2x7

therefore greatest prime factor = 13

Hence D