What is the largest prime factor of the expression 3^8 − 2^12?
A. 2
B. 3
C. 5
D. 17
E. 29
OA E
Source: Veritas Prep
5-Day Free Trial
5-day free, full-access trial TTP
Available with Beat the GMAT members only code
MORE DETAILS
What is the largest prime factor of the expression 3^8 − 2
This topic has expert replies
-
BTGmoderatorDC
- Moderator
- Posts: 7187
- Joined: Thu Sep 07, 2017 4:43 pm
- Followed by:23 members
Timer
00:00
Your Answer
A
B
C
D
E
Global Stats
GMAT/MBA Expert
-
[email protected]
- Elite Legendary Member
- Posts: 10392
- Joined: Sun Jun 23, 2013 6:38 pm
- Location: Palo Alto, CA
- Thanked: 2867 times
- Followed by:511 members
- GMAT Score:800
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
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
GMAT/MBA Expert
-
Brent@GMATPrepNow
- GMAT Instructor
- Posts: 16207
- Joined: Mon Dec 08, 2008 6:26 pm
- Location: Vancouver, BC
- Thanked: 5254 times
- Followed by:1268 members
- GMAT Score:770
There's a difference of squares "hiding" in the expression 3^8 − 2^12BTGmoderatorDC wrote:What is the largest prime factor of the expression 3^8 − 2^12?
A. 2
B. 3
C. 5
D. 17
E. 29
OA E
Source: Veritas Prep
That is 3^8 − 2^12 = (3^4)² − (2^6)²
We know that: x² - y² = (x + y)(x - y)
So, we get: (3^4)² − (2^6)² = (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
-
GMATGuruNY
- GMAT Instructor
- Posts: 15539
- Joined: Tue May 25, 2010 12:04 pm
- Location: New York, NY
- Thanked: 13060 times
- Followed by:1906 members
- GMAT Score:790
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:
x� - y� = (x� + y²)(x� - y²) = (x� + y²)(x² + y¹)(x² - y¹)
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:
x� - y� = (x� + y²)(x� - y²) = (x� + y²)(x² + y¹)(x² - y¹)
Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3
GMAT/MBA Expert
-
Scott@TargetTestPrep
- GMAT Instructor
- Posts: 7247
- Joined: Sat Apr 25, 2015 10:56 am
- Location: Los Angeles, CA
- Thanked: 43 times
- Followed by:29 members
We see that the expression 3^8 - 2^12 is a difference of squares. Thus, we have:BTGmoderatorDC wrote:What is the largest prime factor of the expression 3^8 − 2^12?
A. 2
B. 3
C. 5
D. 17
E. 29
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
Scott Woodbury-Stewart
Founder and CEO
[email protected]
See why Target Test Prep is rated 5 out of 5 stars on BEAT the GMAT. Read our reviews
