Hoping that noone here has ever considered me an "Instigator" (Knewton), I would like to discuss the following problem with you.
GMAT Official Guide Quantitative Review (2nd ed.):
Q 169 Problem Sovling
"If n is a positive integer, and n^2 is divisible by 72, then the largest positive integer that must divide n is:
(A) 6
(B) 12
(C) 24
(D) 36
(E) 48
Correct Answer: B
Official Answer Explanation:
Since n^2 is divisible by 72, n^2 = 72k for some positive integer k. Since n^2 = 72k, then 72k must be a perfect square. Since 72k = (2^3)*(3^2)*k, then k = 2*m^2 for some positive integer m in order for 72k to be a perfect square. Then, n^2 = 72k = (2^3)*(3^2)*(2*m^2) = (2^4)*(3^2)*m^2 = [(2^2)*(3)*(m)]^2, and n = (2^2)*(3)*m. The positive integers, that MUST divide n are 1,2,3,4,6, and 12. Therefore, the largest positive integer that must divide n is 12.
Please discuss:
The boldfaced sentence concerns my question. Why dont we just say that k = 2 so that 72k = 144 which is indeed a perfect square of 12^2 ?? Taking k = 2 instead of 2m^2 would yield the same result namely n^2= [(2^2)*(3)]^2 and thus n = (2^2)*(3) from which we would still be able to conclude that 12 is the largest possible integer that must divide n.
To me, it seemed that the introduction of m^2 did not contribute anything in the finding of the solution. Why did they used it though?
Kind regards,
Tobi
GMAT Official Guide Quantitative Review (2nd ed.):
Q 169 Problem Sovling
"If n is a positive integer, and n^2 is divisible by 72, then the largest positive integer that must divide n is:
(A) 6
(B) 12
(C) 24
(D) 36
(E) 48
Correct Answer: B
Official Answer Explanation:
Since n^2 is divisible by 72, n^2 = 72k for some positive integer k. Since n^2 = 72k, then 72k must be a perfect square. Since 72k = (2^3)*(3^2)*k, then k = 2*m^2 for some positive integer m in order for 72k to be a perfect square. Then, n^2 = 72k = (2^3)*(3^2)*(2*m^2) = (2^4)*(3^2)*m^2 = [(2^2)*(3)*(m)]^2, and n = (2^2)*(3)*m. The positive integers, that MUST divide n are 1,2,3,4,6, and 12. Therefore, the largest positive integer that must divide n is 12.
Please discuss:
The boldfaced sentence concerns my question. Why dont we just say that k = 2 so that 72k = 144 which is indeed a perfect square of 12^2 ?? Taking k = 2 instead of 2m^2 would yield the same result namely n^2= [(2^2)*(3)]^2 and thus n = (2^2)*(3) from which we would still be able to conclude that 12 is the largest possible integer that must divide n.
To me, it seemed that the introduction of m^2 did not contribute anything in the finding of the solution. Why did they used it though?
Kind regards,
Tobi
