What is the Greatest Prime Factor of 6^8−3^8 ? GMAT Problem Solving

Question: What is the greatest prime factor of 6^8−3^8 ?

A) 3
B) 11
C) 17
D) 19
E) 31

“What is the greatest prime factor of 6^8−3^8 ?" - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been borrowed from the book “GMAT Official Guide Quantitative Review”.

To understand GMAT Problem Solving questions, applicants must possess fundamental qualitative skills. Quant tests a candidate's aptitude in reasoning and mathematics. The GMAT Quantitative test's problem-solving phase consists of a question and a list of possible responses. By using mathematics to answer the question, the candidate must select the appropriate response. The problem-solving section of the GMAT Quant topic is made up of very complicated math problems that must be solved by using the right math facts.

Answer: C
Solution and Explanation:
Approach Solution 1:

6⁸ − 3⁸ is a DIFFERENCE OF SQUARES. So we can factor it.
6⁸ − 3⁸ = (6⁴ + 3⁴)(6⁴ - 3⁴)
= (6⁴ + 3⁴)(6² + 3²)(6² - 3²)
= (6⁴ + 3⁴)(6² + 3²)(6 + 3)(6 - 3)
= (6⁴ + 3⁴)(45)(9)(3)
= (6⁴ + 3⁴)(3)(3)(5)(3)(3)(3)

Hmmmm, we can see that the correct answer is "hiding" in the first number (6⁴ + 3⁴)

Let's factor out the 3⁴, to get:

6⁴ + 3⁴ = 3⁴(2⁴ + 1)
= 3⁴(16 + 1)
= 3⁴(17)
= (3)(3)(3)(3)(17)

So, 6⁸ − 3⁸ = (3)(3)(3)(3)(17)(3)(3)(5)(3)(3)(3)

So the correct answer is C

Correct Answer: C

Approach Solution 2:

It is asked to find out the greatest prime factor of 6^8 - 3^8.
6^8 − 3^8
2^8∗3^8−3^8
3^8(2^8−1)
(2^4+1)(2^4−1)
(16+1)(16−1)
(17)(15)
So 17 is Answer

Hence, C is the correct answer.

Correct Answer: C

Approach Solution 3:

First figure out the common factors in both
6868 can be written s 2^8∗3^8
Take 3^8 common

3^8∗2^8−3^8
3^8(2^8−1) (here you should know that 2^8=256, The easiest way to remember this is 2^10 = 1024 and now you can derive most 2x)
SO your expression becomes 3^8(256−1) ===> 3^8(255)==>3^8(17∗15)
so your prime factorisation will be 3^8∗15∗17===>3^8∗(3^1∗5^1)∗17=====>3^9∗5^1∗17
So as we can see, there are three prime numbers here 3, 5, and 17, out of which 3 is the smallest prime number and 17 is the biggest.

Hence, 17 is the biggest prime factor

Correct Answer: C

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