Lesson: Squares of Integers

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Hi Brent,

Could you please help solve this question using your methods?

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
gmat-admin's picture

You bet!
You'll find my step-by-step solution here: https://gmatclub.com/forum/if-n-is-a-positive-integer-and-n-2-is-divisib...

Cheers,
Brent

Hi Brent,

Regarding this question: How many positive divisors of 12,500 are squares of integers (aka perfect squares)?

A) three
B) four
C) six
D) eight
E) twelve

Is my following deduction true, or was I simply lucky to obtain the right answer?

My thinking below:
We can find the Prime Factors of 12 500 which are 5^5 and 2^2.

The question asks to obtain the number of positive, perfect square divisors of 12, 500. To do so, we must apply the rule that "The Prime Factorization of a perfect square will have an even number of each prime".

Given this rule, we can see that the PFs of 12 ,500 = (5^5) (2^2). However, it is clear that 5^5 is ODD, which means that the number of positive, perfect square divisors of 12, 500 must be even. Therefore we can infer that the correct answer must be the next best EVEN pair of Prime Factors, that being: (5^4) (2^2).

This gives us a total of 6 numbers if we add the exponents.

Appreciate the help!
gmat-admin's picture

Unfortunately, it was only a coincidence that adding the exponents of (5^4)(2^2) yielded the correct answer.

Notice that, if we were to determine how many divisors of (5^4) are squares of integers, we'd conclude (using your approach) that there are 4 such divisors, while there are actually only 3.

Likewise, if we were to determine how many divisors of (2^2)(3^2)(5^2) are squares of integers, we'd conclude (using your approach) that there are only 6 such divisors, while there are actually 8.

Cheers,
Brent

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