What is the largest prime factor of the expression 3^8 âˆ’ 2

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BTGmoderatorDC: Moderator; Posts: 7187; Joined: Thu Sep 07, 2017 4:43 pmFollowed by: 23 members

by BTGmoderatorDC » Thu Oct 25, 2018 5:18 pm

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What is the largest prime factor of the expression 3^8 âˆ’ 2^12?

A. 2
B. 3
C. 5
D. 17
E. 29

OA E

Source: Veritas Prep

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Source: Beat The GMAT — Problem Solving | Thu Oct 25, 2018 5:18 pm

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by [email protected] » Thu Oct 25, 2018 6:43 pm

Hi All,

We're asked to find the LARGEST prime factor of the expression 3^8 - 2^12. Certain Quant questions ultimately come down to 'rewriting' information that you've been given. By rewriting 3^8 and 2^12 as "squared" terms, we'll have a 'difference of squares', which is a Classic Quadratic pattern in Algebra.

3^8 = 9^4 = 81^2
2^12 = 4^6 = 64^2

Thus, we have...
81^2 - 64^2 =
(81 - 64)(81 + 64) =
(17)(145) =
(17)(5)(29)

The largest prime factor is 29.

Final Answer: E

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

Contact Rich at [email protected]

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Fri Oct 26, 2018 5:34 am

BTGmoderatorDC wrote:What is the largest prime factor of the expression 3^8 âˆ’ 2^12?

A. 2
B. 3
C. 5
D. 17
E. 29

OA E

Source: Veritas Prep

There's a difference of squares "hiding" in the expression 3^8 âˆ’ 2^12
That is 3^8 âˆ’ 2^12 = (3^4)Â² âˆ’ (2^6)Â²

We know that: xÂ² - yÂ² = (x + y)(x - y)

So, we get: (3^4)Â² âˆ’ (2^6)Â² = (3^4 + 2^6)(3^4 - 2^6)
= (81 + 64)(81 - 64)
= (145)(17)
= (5)(29)(17)

The greatest prime factor is 29

Answer: E

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Fri Oct 26, 2018 6:00 am

A general rule:
Given x^(even power) - y^(even power), where x and y are distinct positive integers, we can always simplify by applying the difference of two squares.
Each time we apply the difference of two squares to the expression in blue, the exponents are reduced by 1/2.
For example:
xâ�¸ - yâ�´ = (xâ�´ + yÂ²)(xâ�´ - yÂ²) = (xâ�´ + yÂ²)(xÂ² + yÂ¹)(xÂ² - yÂ¹)

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by Scott@TargetTestPrep » Sat Nov 03, 2018 4:59 pm

BTGmoderatorDC wrote:What is the largest prime factor of the expression 3^8 âˆ’ 2^12?

A. 2
B. 3
C. 5
D. 17
E. 29

We see that the expression 3^8 - 2^12 is a difference of squares. Thus, we have:

3^8 - 2^12 = (3^4 + 2^6)(3^4 - 2^6) = (3^4 + 2^6)(3^2 + 2^3)(3^2 - 2^3)

3^4 + 2^6 = 81 + 64 = 145 = 5 x 29

3^2 + 2^3 = 9 + 8 = 17

3^2 - 2^3 = 9 - 8 = 1

So 3^8 - 2^12 = 5 x 29 x 17, and the largest prime factor is 29.

Answer: E

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Founder and CEO
[email protected]

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