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## Comment on

Squares of Integers## 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

## You bet!

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!

## Unfortunately, it was only a

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

## Hi Brent, need help with this

## I'm happy to help!

I'm happy to help!

Here's my full solution: https://gmatclub.com/forum/if-n-is-a-positive-integer-and-n-2-is-divisib...

Cheers,

Brent

## "A perfect square will have

The converse must be true as well?

## Great question! Yes, the

Great question! Yes, the converse is true.

That is: If a positive integer N has an odd number of positive divisors, then N is a perfect square.

## Great question! Yes, the

Great question! Yes, the converse is true.

That is: If a positive integer N has an odd number of positive divisors, then N is a perfect square.

## Hi Brent,

Is it also a true statement:

If all prime factors(when the same primes combined together 2x2=2^2) of a number X have even powers then the number is a perfect square?

Thank you in advance,

## That's also true! Nice work!

That's also true! Nice work!

Let's formalize your observation:

If, in the prime factorization of N, all exponents are even, then N is a perfect square.

## Hi Brent, could you please

The number 75 can be written as the sum of the squares of 3 different positive integers. What is the sum of these 3 integers?

A.17

B.16

C.15

D.14

E.13

Thanks!

## The tricky thing about this

The tricky thing about this question is that there isn't a nice algebraic solution.

It all comes down to finding 3 values (from the numbers 1, 4, 9, 16, 25, 36, 49, and 64) that add to 75.

Here's my solution: https://www.beatthegmat.com/the-number-75-can-be-written-as-the-sum-of-t...

Cheers,

Brent

## Is w greater than 1 ?

(1) w + 2 > 0

(2) w² > 1

Shouldnt the answer be C?

If we combine both statements the value of w will be greater than 2.

## Statement 1 tells us that w >

Statement 1 tells us that w > -2

Statement 2 tells us that EITHER w < -1 OR w > 1

When we combine the statements, it COULD be the case that:

case i: w = -1.5, in which case x is NOT greater than 1

case ii: w = 8, in which case x IS greater than 1

Answer: E

Does that help?

Cheers,

Brent

## Hi Brent,

Could you help me with this problem:

For every positive even integer n, the function h(n) is defined to be the product of all the even integers from 2 to n, inclusive. If p is the smallest prime factor of h(100) + 1, then p is

A)between 2 and 10

B)between 10 and 20

C)between 20 and 30

D)between 30 and 40

E)greater than 40

## Hi sandab,

Hi sandab,

Here's my approach: https://gmatclub.com/forum/for-every-positive-even-integer-n-the-functio...

Cheers,

Brent

## Hey Brent,

so there exists absolutely NO number that has an odd nr. of divisors and i NOT a perfect square? Means: If a nr. has an odd nr. of divisors it has to be a perfect square, no way around it?

Cheers,

Philipp

## That's correct.

That's correct.

If integer N has an odd number of positive divisors, then N is a perfect square.

If integer N has an even number of positive divisors, then N is not a perfect square.

## https://gmatclub.com/forum/if

Could you help me in this regard?

Thanks

## Here's my full solution:

Here's my full solution: https://gmatclub.com/forum/if-the-integer-n-has-exactly-three-positive-d...

Cheers,

Brent

## Hey Brent,

How were you able to quickly deduce the need to start at 30. I knew immediately some where squares but not all.

https://gmatclub.com/forum/what-is-the-largest-3-digit-number-to-have-an-odd-number-of-factors-214645.html

## Solution link: https:/

Solution link: https://gmatclub.com/forum/what-is-the-largest-3-digit-number-to-have-an...

We're looking for the biggest 3-digit number that's also the square of an integer.

I recognized that 30² is one of the bigger 3-digit numbers. So, I started there.