For p^mq^n, the number of divisors is (m+1)(n+1).

Thus the number of divisors is (3+1)(6+1) = 28

The answer is D) 28.

prime numbers only. if you have 6^2, make it 2^2 and 3^2 => therefore you have (2+1)(2+1)=9 factors

mrblack and jade, correct me if i am wrong

The calculate the number of divisors of 36 -----(2^2 * 3^2) = (2+1)(2+1) = 9

Similarly to calculate for 36 * 48 ------(2^2 * 3^2) and (2^4 * 3) = (2+1)(2+1)*(4+1)(3+1) =180

Yes 36 has 9 divisors

But 36*48 does not have 180 divisors(The colored step is wrong. reduce every thing to prime factors before proceeding)

36*48 is (2^2*3^2)*(2^4*3) = 2^6*3^3

So the number of divisors is (6+1)*(3+1)=28

The below is the list of all 28 divisors of 36*48
1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 96, 108, 144, 192, 216, 288, 432, 576, 864, 1728

The method of finding the number of factors (also called divisors) is based on the concept of finding the basic factors (which make up all other factors) and then combining them in different ways to make as many different factors as possible.

Let us say the question asks us to find the number of factors of 72.
We know that 72 = 8x9 = 2^3 x 3^2 - This is called prime factorization. We have essentially brought down 72 to its basic factors.
We find that 72 has three 2s and two 3s. We can combine them in various ways e.g. I could take one 2 and two 3s and make 2x3x3 = 18. Similarly, I could take three 2s and no 3 to make 2x2x2 = 8
Since we have three 2s, we can choose a 2 in four ways (take no 2, take one 2, take two 2s or take three 2s)
Since we have two 3s, we can choose a 3 in three ways (take no 3, take one 3 or take two 3s)
Every time we make a different choice, we get a different factor of 72.
Since we can choose 2s in 4 ways and 3s in 3 ways, together we can choose them in 4x3 = 12 ways. Therefore, 72 will have 12 factors.

This is true for any positive integer N.
If N = 36 x 48 = 2^2 x 3^2 x 2^4 x 3 = 2^6 x 3^3.
To make factors of N, we have seven ways to choose a 2 and four ways to choose a 3. Therefore, we can make 7 x 4 = 28 combinations of these prime factors to give 28 factors of 36x48.
Note: I could write 36 x 48 as 72 x 24 or 64 x 27 or 8 x 8 x 27 or many other ways. The answer doesn't change because it is still the same number N.

Generalizing, if N = p^a x q^b x r^c ... where p, q, r are all distinct prime numbers, the total number of factors of N (including 1 and N ) is (a + 1)(b + 1) (c + 1)...
Remember, the '+1' is because of an option of dropping that particular prime number from our factor.

On that note, if I tell you that a positive integer N has total 7 factors, what can you say about N?

swaraj and VeritasPrepKarishma, thank you. I got it.

thank you... i understood how to solve it....

To determine the number of positive factors of any integer:

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

A question in OG11 (I think) asked for the number of positive factors of 441.

Since 441 = 3^2 * 7^2, we get (2+1)(2+1) = 9 factors.

Here's the reasoning. To determine how many factors can be created from 3^2 * 7^2, we need to determine the number of choices we have of each prime factor:

For 3, we can use 3^0, 3^1, or 3^2, giving us 3 choices.
For 7, we can use 7^0, 7^1, or 7^2, giving us 3 choices.

Multiplying, we get 3*3 = 9 possible factors.
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When you need to find the number of divisors of a number, you use this approach: Break down the number into its prime factors. e.g. N = a^p*b^q*c^r... where a, b and c are all distinct prime factors of N. p, q and r are the powers of the prime factors in N
Total number of divisors of N = (p+1)(q+1)(r+1)...

e.g. Total number of factors of 36 (= 2^2*3^2) is (2+1)(2+1) = 9

The detailed theory for this has been given here:
http://www.veritasprep.com/blog/2010/12/quarter-wit-quarter-wisdom-writing-factors-of-an-ugly-number/

In this question, you need to find the total number of divisors of (P^3)(Q^6) where P and Q are prime.
Total number of factors = (3+1)(6+1) = 28

Note: They should have mentioned that P and Q are distinct prime numbers. If P and Q are not distinct e.g. 3^3*3^6 = 3^9 and its total number of divisors is (9+1) = 10. But from the options, it is obvious they intend you to take them as distinct. Still, erroneous question.
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