There is a question which has an explanation that I am not entirely clear on.
Question :
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
Explanation :
Answer is 12. 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 = 2m^2 for some positive integer m in order for 72k to be a perfect square. Then, n^2 = 72k = (2^3)(3^2)(2m^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 answer is 12.
The part I get confused on is where it talks about k=2m^2 for some positive integer m in order for 72k to be a perfect square. I am fairly sure there is a simpler or more straight forward explanation and am hoping someone can help me clear this up. Or what kind of mindset should I be going with when solving this kind of problem.
Much appreciated in advance!
Question :
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
Explanation :
Answer is 12. 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 = 2m^2 for some positive integer m in order for 72k to be a perfect square. Then, n^2 = 72k = (2^3)(3^2)(2m^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 answer is 12.
The part I get confused on is where it talks about k=2m^2 for some positive integer m in order for 72k to be a perfect square. I am fairly sure there is a simpler or more straight forward explanation and am hoping someone can help me clear this up. Or what kind of mindset should I be going with when solving this kind of problem.
Much appreciated in advance!
