Gmat loves factors

This topic has expert replies
User avatar
Legendary Member
Posts: 1079
Joined: 13 Dec 2010
Thanked: 118 times
Followed by:33 members
GMAT Score:710

Gmat loves factors

by bblast » Sun Jan 09, 2011 4:27 am
In the number 36 :

1>How many total factors ?
2>How many Odd factors ?
3>How many even factors ?
4>How many prime factors ?

This is a self devised question, so no OA.

BTW IMO answers are
[spoiler]1>9
2>3
3>6
4>2[/spoiler]

Short cuts after expert opinions.
Cheers !!

Quant 47-Striving for 50
Verbal 34-Striving for 40

My gmat journey :
https://www.beatthegmat.com/710-bblast-s ... 90735.html
My take on the GMAT RC :
https://www.beatthegmat.com/ways-to-bbla ... 90808.html
How to prepare before your MBA:
https://www.youtube.com/watch?v=upz46D7 ... TWBZF14TKW_

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 3835
Joined: 02 Apr 2010
Location: Milpitas, CA
Thanked: 1854 times
Followed by:523 members
GMAT Score:770

by [email protected] » Sun Jan 09, 2011 4:39 am
bblast wrote:In the number 36 :

1>How many total factors ?
2>How many Odd factors ?
3>How many even factors ?
4>How many prime factors ?
36 = (2^2)*(3^2)

1. Number of total factors = (Number of ways to select any number of 2's out of 2)*(Number of ways to select any number of 3's out of 2) = (2 + 1)*(2 + 1) = 9

2. Number of total odd factors = (Number of ways to select no 2)*(Number of ways to select any number of 3's out of 2) = 1*(2 + 1) = 3

3. Number of even factors = (Number of ways to select at least one 2 out of 2)*(Number of ways to select any number of 3's out of 2) = (2)*(2 + 1) = 6

4. Number of prime factors = 2
Anurag Mairal, Ph.D., MBA
GMAT Expert, Admissions and Career Guidance
Gurome, Inc.
1-800-566-4043 (USA)

Join Our Facebook Groups
GMAT with Gurome
https://www.facebook.com/groups/272466352793633/
Admissions with Gurome
https://www.facebook.com/groups/461459690536574/
Career Advising with Gurome
https://www.facebook.com/groups/360435787349781/

Master | Next Rank: 500 Posts
Posts: 131
Joined: 18 Jun 2010
Location: New York, NY
Thanked: 10 times

by aleph777 » Mon Jan 10, 2011 11:12 am
[email protected],

I'm not familiar with your method of breaking down total factors. Can you explain in a bit more detail?

Thanks!

User avatar
GMAT Instructor
Posts: 15533
Joined: 25 May 2010
Location: New York, NY
Thanked: 13060 times
Followed by:1900 members
GMAT Score:790

by GMATGuruNY » Mon Jan 10, 2011 11:46 am
In the number 36 :

1>How many total factors ?
2>How many Odd factors ?
3>How many even factors ?
4>How many prime factors ?

This is a self devised question, so no OA.

BTW IMO answers are
[spoiler]1>9
2>3
3>6
4>2[/spoiler]
To determine the number of positive factors of an integer:

1) Prime-factorize the integer
2) Add 1 to each exponent
3) Multiply


36 = 2^2 * 3^2. Adding 1 to each exponent and multiplying, we get (2+1)*(2+1) = 9 factors.

Here's the reasoning. To determine how many factors can be created from 36 = 2^2 * 3^2, we need to determine the number of choices we have of each prime factor:

For 2, we can use 2^0, 2^1, or 2^2, giving us 3 choices.
For 3, we can use 3^0, 3^1, or 3^2, giving us 3 choices.

Multiplying, we get 3*3 = 9 possible factors.

Another example: How many positive factors does 882 have?

882 = 2 * 3^2 * 7^2. Adding 1 to each exponent and multiplying, we get 2*3*3 = 18 factors.

To determine the number of odd positive factors of an integer:

1) Prime-factorize the integer
2) Add 1 to the exponent of each odd prime factor
3) Multiply


36 = 2^2 * 3^2. The only odd prime factor is 3, with an exponent of 2. Adding 1 to the exponent, we get 2+1 = 3 odd factors.

Number of even positive factors = Total possible factors - Odd factors = 9-3 = 6.
Mitch Hunt
Private Tutor for the GMAT and GRE
[email protected]

If you find one of my posts helpful, please take a moment to click on the "UPVOTE" icon.

Available for tutoring in NYC and long-distance.
For more information, please email me at [email protected].
Student Review #1
Student Review #2
Student Review #3

User avatar
Legendary Member
Posts: 502
Joined: 03 Jun 2008
Thanked: 99 times
Followed by:21 members

by vk_vinayak » Mon Sep 24, 2012 3:36 am
So, to calculate EVEN POSITIVE FACTORS, we must find the total factors and subtract ODD POSITIVE FACTORS from it?

From 36 = 2^2 * 3^2. Why can't we say that only EVEN prime factor is 2, with an exponent of 2 and adding 1 to the exponent, we get 2+1 = 3 EVEN factors ?
- VK

I will (Learn. Recognize. Apply)

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 15735
Joined: 08 Dec 2008
Location: Vancouver, BC
Thanked: 5254 times
Followed by:1266 members
GMAT Score:770

by [email protected] » Mon Sep 24, 2012 6:31 am
aleph777 wrote: I'm not familiar with your method of breaking down total factors. Can you explain in a bit more detail?
If N = (p^a)(q^b)(r^c)..., where p, q, r,...(etc.) are prime numbers, then the total number of positive divisors of N is equal to (a+1)(b+1)(c+1)...

Example: 14000 = (2^4)(5^3)(7^1)
So, the number of positive divisors of 14000 = (4+1)(3+1)(1+1) = 5x4x2=40

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
Image

User avatar
GMAT Instructor
Posts: 15533
Joined: 25 May 2010
Location: New York, NY
Thanked: 13060 times
Followed by:1900 members
GMAT Score:790

by GMATGuruNY » Mon Sep 24, 2012 11:49 am
vk_vinayak wrote:So, to calculate EVEN POSITIVE FACTORS, we must find the total factors and subtract ODD POSITIVE FACTORS from it?

From 36 = 2^2 * 3^2. Why can't we say that only EVEN prime factor is 2, with an exponent of 2 and adding 1 to the exponent, we get 2+1 = 3 EVEN factors ?
This approach counts one combination that is NOT even (2�) but omits many combinations that ARE even (2*3, 2*3², etc.).
A factor will be EVEN if its prime-factorization includes AT LEAST ONE 2.
To directly count the EVEN positive factors of a positive integer, we could do the following:

1. Prime-factorize the integer.
2. Add 1 to every exponent OTHER THAN 2's exponent.
3. Multiply the results by 2's exponent.


To illustrate:
720 = 2� * 3² * 5¹
The total number of EVEN factors = (4)(2+1)(1+1) = 24.

The reason that we DON'T add 1 to 2's exponent is that an EVEN factor must include AT LEAST ONE 2, so 2� is not an option.
An even factor of 720 must include either 2¹, 2², 2³, or 2�.
Thus, the total number of options with regard to 2� is 4 -- the value of 2's exponent.
Mitch Hunt
Private Tutor for the GMAT and GRE
[email protected]

If you find one of my posts helpful, please take a moment to click on the "UPVOTE" icon.

Available for tutoring in NYC and long-distance.
For more information, please email me at [email protected].
Student Review #1
Student Review #2
Student Review #3

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 2612
Joined: 02 Jun 2008
Location: Toronto
Thanked: 1090 times
Followed by:355 members
GMAT Score:780

by Ian Stewart » Mon Sep 24, 2012 8:45 pm
If you have the prime factorization of an even number, and it looks like this:

(2^k) * some odd primes

then the ratio of the number of even factors to the number of odd factors is k to 1.

So if you take a number like:

120 = (2^3)(3)(5)

then the ratio of even to odd divisors is 3 to 1, and so 3/4 of the factors of 120 will be even, and 1/4 of the factors of 120 will be odd.

As a consequence, every even number has at least as many even divisors as odd divisors, and any multiple of 2^2 = 4 has at least twice as many even divisors as odd divisors.
If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com