What is the greatest prime factor of 42^2+75^2+6300 ? (A) 3

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hazelnut01: Master | Next Rank: 500 Posts; Posts: 157; Joined: Sat Nov 19, 2016 5:34 amFollowed by: 4 members; Thanked: 2 times

by hazelnut01 » Fri May 12, 2017 12:10 am

What is the greatest prime factor of 42^2+75^2+6300 ?

(A) 3
(B) 7
(C) 11
(D) 13
(E) 79

OA=D

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Source: Beat The GMAT — Problem Solving | Fri May 12, 2017 12:10 am

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 May 12, 2017 3:22 am

ziyuenlau wrote:What is the greatest prime factor of 42^2+75^2+6300 ?

(A) 3
(B) 7
(C) 11
(D) 13
(E) 79

The given expression looks like the red portion of the following:
(a+b)Â² = aÂ² + bÂ² + 2ab.

Test where (42 + 75)Â² = 42Â² + 75Â² + 6300:
(42 + 75)Â²

= 42Â² + 75Â² + 2*42*75

= 42Â² + 75Â² + 2*6*7*5*5*3

= 42Â² + 75Â² + 10*30*21

= 42Â² + 75Â² + 6300.
Success!

Thus:
42Â² + 75Â² + 6300 = (42+75)Â² = 117Â².
Of the five answer choices, the greatest value that divides evenly into 117 is D:
117 = 13*9.

The correct answer is D.

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8176; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

Re: What is the greatest prime factor of 42^2+75^2+6300 ? (A) 3

by Scott@TargetTestPrep » Mon Apr 13, 2020 3:50 am

hazelnut01 wrote: ↑
Fri May 12, 2017 12:10 am
What is the greatest prime factor of 42^2+75^2+6300 ?

(A) 3
(B) 7
(C) 11
(D) 13
(E) 79

OA=D

Consider the perfect square (a + b)^2, which expands to a^2 + b^2 + 2ab.

This pattern can be applied to (42 + 75)^2, which expands to 42^2 + 75^2 + 2(42)(75). Note that the third term of this expansion 2(42)(75) = 6300. Thus, we see that 42^2 + 75^2 + 6300 = (42 + 75)^2. Now, perform the addition inside the parentheses, obtaining 117^2 = (3^2 x 13)^2. We see that the greatest prime factor is 13.

Answer: D

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