FIRST:

Finding the Number of Factors of an Integer

First make prime factorization of an integer n=a^p*b^q*c^r, where a, b, and c are prime factors of n and p, q, and r are their powers.

The number of factors of n will be expressed by the formula (p+1)(q+1)(r+1). NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: 450=2^1*3^2*5^2

Total number of factors of 450 including 1 and 450 itself is (1+1)*(2+1)*(2+1)=2*3*3=18 factors.

BACK TO THE ORIGINAL QUESTION:

According to the above as 72=2^3*3^2, then # of factors of 72 is (3+1)(2+1)=12. Out of which only 3 are odd 1, 3, and 9, so rest or 12-3=9 are even.

OR: as 72=2^3*3^2 then even factors MUST have 2 either in power of 1, 2, or 3 so 3 options and 3 either in power 0, 1, or 2 again 3 options --> 3*3=9.

Answer: E.

Hope it helps.
_________________

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. NEW!!!

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set. NEW!!!


What are GMAT Club Tests?
25 extra-hard Quant Tests

Find out what's new at GMAT Club - latest features and updates

you were lucky there I'm afraid the method is incorrect.

Sol:

The number of factors of 72 will be 12 and not 10. The best way to find out is if X= a^b * c^d then number of factors are (b+1) * (d+1)

here 72= 2^3*3^2 so number of factors will be (3+1) * (2+1) = 12 --> this includes 1 and the number itself.

so for finding factors not divisible by 2, remove all the 2s from the prime factorization. You will be left with 3^2.

so factors will be 3 and 3^2(=9). Also, we need to include the number 1 to this list, as it is odd and not divisible by 2.

so 12-3=9

the flaw in your approach:
1.) 5!/2!*3! will give you the number of ways you can arrange (permute) 22233. essentially it gives you the following list:
22233,22323,33222,32322 etc. As you can see this is a mere representation of how you can write three 2s and two 3s. This does NOT give you the list of factors of 72.
2.) not only 3*3 is odd but 3 and 3*3 both are odd factors. Include 1 to this list and you get the 3 odd factors.


hope this helps.
_________________

My Debrief | Find out what's new at GMAT Club

BTW - jumsumtak actually shows you how to find the number of factors for a number, and will be a time saver in the exam. Note the example here works with only two prime factors. For another example with three prime factors: "How many factors in 360?", 360 = 5 x 8 x 9 --> (1+1) x (3+1) x (2+1) = 24.

Merging similar topics. Please refer to the solutions above.
_________________

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. NEW!!!

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set. NEW!!!


What are GMAT Club Tests?
25 extra-hard Quant Tests

Find out what's new at GMAT Club - latest features and updates

(Will try to explain this using an easier but slightly slower approach since many have difficulties grasping perms & combs)

Using the number 72 from the original question;
[1] Find number of factors:
1. 72 = 2^3 x 3^2;
2. therefore number of factors = (3+1) x (2+1) = 12

[2] Find number of ODD factors:
*Number property, N1: We know that all primes, except 2, are odd.
*Number property, N2: We know that ODD x ODD = ODD.
*Number property, N3: Multiplying any number by 2 (an Even Number) will yield an EVEN number.

1. Recalling from [1], we have identified 2 and 3 as the prime factors of 72.
2. We ignore the "2" remembering N3.
3. We can construct factors that consist of prime factor 3 only:3, 3 x 3 (since there are only two "3"s we stop here).
4. Let's not forget that "1" is also a non-even factor.
5. Total of ODD factors is 3.

[3] Find number of EVEN factors: Total of factors - total of odd factors = total of even factors = 12 - 3 = 9.


One could directly use combinations of 2, 2, 2, 3, 3 to list all EVEN factors but I've found it faster to find ODD factors first.

For example, in my explanation for counting factors for a number with three distinct primes:
1. 360 = 5 x 8 x 9 = 5^1 x 2^3 x 3^2
2. Number of primes = (1+1) x (3+1) x (2+1) = 24
3. Number of odd factors will be multiples of only up to 1 "5" and 2 "3"s.
4. List odd factors:
3
5
9 = 3 x 3
15 = 3 x 5
45 = 3 x 3 x 5
5. Do not forget that "1" is also a factor, therefore there are 6 ODD factors in 360.
6. Total of EVEN factors in 360 is 24 - 6 = 18.

*Quick Check with pairs indeed reveals 6 ODD factors:
1, 360 --> 1 is ODD
2, 180
3, 120 --> 3 is ODD
4, 90
5, 72 --> 5 is ODD
6, 60
8, 45 --> 45 is ODD
9, 40 --> 9 is ODD
10, 36
12, 30
15, 24 --> 15 is ODD
18, 20


Hope this clarifies things.
* Do note that I would expect 750+ questions to involve combinatorics that involve the use of perms & combs to solve within the time limit.