Perfect square

Question

The number theory states that : A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. Could you please explain what this means :?:

If i have a number say: 4624.that will be 17*17*2*2*2*2 which is 2 factors of the odd number 17 :roll:

EliLTG · Answer

Hi Motivated,

Interesting trivia! As a mathematician, I'm always interested to learn facts like that--but as a GMAT instructor, I always caution students about focusing too much on details like this that are  extremely  unlikely to appear on your exam!

As for your specific question, you are getting factors confused with  prime  factors. The rule you mentioned applies to the total list of factors of 4624, i.e. 1 * 4624, 2 * 2312, 4 * 1156...

Hope this helps!

Motivatedtowin · Answer

Thank you so much ...The exam already has too much to study and work in itself :shock:

But i just came across this and could not understand..So basically..say for a smaller number like 36:1*2*2*3*3, number of odd factors=3(ODD) and number of even factors equals 2(EVEN) :-D

EliLTG · Answer

Not quite! 
The factors of 36 are:

1, 36
2, 18
3, 12
4, 9
6, 6

So, the odd factors are: 1, 3, and 9;
The even factors are 2, 4, 6, 12, 18, 36.
Does this help?

KarishmaB · Answer

Here are a few interesting things about factors of perfect squares:

ONLY perfect squares will have odd number of total factors and ALL perfect squares will have an odd number of total factors.

A perfect square always has odd number of odd factors.

A perfect square always has even number of even factors.

The sum of all factors of a perfect square is always odd but if the sum of all factors of a number is odd, we cannot say that it must be a perfect square.

Motivatedtowin · Answer

Thanks so much for the replies...Finally clarity

woohoo921 · Answer

KarishmaB 
Hi my friend! Unfortunately, I could not access the link that you shared. Would you be able to share more information on this form?

I was a bit confused because I learned "an odd number of factors=a perfect square".

So, if I take the square root of 4 --> 4^(1/2)=2*2 -->2 --> because there is only one integer, that makes a perfect square?

Thank you for all of your time.

NEYR0N · Answer

KarishmaB  , you mean

A perfect square always has odd number of DISTINCT odd factors.

A perfect square always has even number of DISTINCT even factors.

is this all we need for perfect squares theory in GMAT?

thanks,

Thanks in advance,

KarishmaB · Answer

Yes, total number of factors implies distinct total factors. 
And no, there is no such exhaustive list. You must understand the what and why of factors and factorization so that whatever question you get, you are able to solve it.

Perfect square

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