It is currently 23 Oct 2017, 10:43

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# GMAT Number properties - perfect squares

Author Message
TAGS:

### Hide Tags

Intern
Joined: 01 Mar 2011
Posts: 8

Kudos [?]: 8 [1], given: 1

GMAT Number properties - perfect squares [#permalink]

### Show Tags

05 May 2011, 19:27
1
KUDOS
5
This post was
BOOKMARKED
Hello Brains,
I recently made a few points about perfect squares that might be helpful to solve problems quickly.

These are the props :
A perfect square has an even number of powers of prime factors
Any perfect square integer always has an odd number of distinct factors.
For any perfect square, the sum of its distinct factors is always odd
Now for the substantiations:
A perfect square has an even number of powers of prime factors
This is "THE BASIC RULE" and the other two (and perhaps many others) rules can be derived out of this rule. The rule says - For a perfect square, N, if N is prime factorized, say N = (px) * (qy), x and y can and will only be even integers. This seems pretty obvious. If x and/or y were infact odd, there wouldn't be able to find sqrt(N) in an even positive integer.

With that now settled, in order to prove the second point,

Any perfect square integer always has an odd number of distinct factors.
there is this tiny hack that lets you find the number of distinct factors a number has. it is a simple two step process.
Factor the number into its prime components, N = (px) * (qy)
The number of distinct factors =(x+1)*(y+1).
Since we proved just now that for such a perfect square x and y will be even,
Code:
the number of factors = (even + 1)*(even + 1)
= odd* odd
= odd

The third rule is a bit tricky

For any perfect square, the sum of its distinct factors is always odd
I havent arrived at a thorough proof for this one as of now. But one thing to remember is that any perfect square will have an odd number of odd factors and an even number of even factors. So adding all these up, we get an odd integer. Try it.

Kudos [?]: 8 [1], given: 1

VP
Status: There is always something new !!
Affiliations: PMI,QAI Global,eXampleCG
Joined: 08 May 2009
Posts: 1285

Kudos [?]: 283 [0], given: 10

Re: GMAT Number properties - perfect squares [#permalink]

### Show Tags

08 May 2011, 00:37
May be this snippet from the GMAT Club Math Book wil help you deduce the proof.

math-number-theory-88376.html

Finding the Number of Factors of an Integer

First make prime factorization of an integer , where , , and are prime factors of and , , and are their powers.

The number of factors of will be expressed by the formula . NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450:

Total number of factors of 450 including 1 and 450 itself is factors.

Finding the Sum of the Factors of an Integer

First make prime factorization of an integer , where , , and are prime factors of and , , and are their powers.

The sum of factors of will be expressed by the formula:

Example: Finding the sum of all factors of 450:

The sum of all factors of 450 is
_________________

Visit -- http://www.sustainable-sphere.com/
Promote Green Business,Sustainable Living and Green Earth !!

Kudos [?]: 283 [0], given: 10

GMAT Club Legend
Joined: 09 Sep 2013
Posts: 16534

Kudos [?]: 274 [0], given: 0

Re: GMAT Number properties - perfect squares [#permalink]

### Show Tags

27 Sep 2015, 07:40
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________

Kudos [?]: 274 [0], given: 0

GMAT Club Legend
Joined: 09 Sep 2013
Posts: 16534

Kudos [?]: 274 [0], given: 0

Re: GMAT Number properties - perfect squares [#permalink]

### Show Tags

21 Dec 2016, 19:22
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________

Kudos [?]: 274 [0], given: 0

Re: GMAT Number properties - perfect squares   [#permalink] 21 Dec 2016, 19:22
Display posts from previous: Sort by