If p and q are prime numbers, how many divisors does the pro

Question

If p and q are prime numbers, how many divisors does the product p^3*q^6 have?

(A) 9 
(B) 12
(C) 18
(D) 28
(E) 363

Bunuel · Accepted Answer

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,  p^3*q^6  ( assuming p and q are different primes ) will have  (3+1)(6+1)=28  different positive factors.

Answer: D.

Hope it helps.

KarishmaB · Answer

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?

KarishmaB · Answer

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

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.

BassanioGratiano · Answer

I went another route that is probably more complicated.

I went to go figure out the number of combinations you can put

p, p^2, p^3 and q, q^1, q^2.... q^6 together

Essentially there are 18 different combinations you can make (3x6)

Also you have to include the different combinations the p and q exponents can be a divisor. There are 3 different p exponents and 6 different q exponents: p, p^2, p^3, q, q^2, q^3... q^6

Also add 1 since "1" can also be a divisor.

18+3+6+1 = 28

philipssonicare · Answer

Added 'poor quality' tag as it is not made clear that they are different primes (though only one fits in the answer choice).

p^3*q^6 
 2^3·3^6=(1+3)(1+6)=28 
 2^3·2^6=2^9=(1+9)=10

BrushMyQuant · Answer

Given that p and q are prime numbers and we need to find the number of divisors of  p^3*q^6

Number of divisors of a number written in the form  p^3*q^6 , where p and q are prime numbers is given by the following theory

Theory: To find number of factors of a number we need to write the number as product of power of prime number and add one to the powers and multiply the powers

=> (3+1)*(6+1) = 4*7 = 28

So,  Answer will be D 
Hope it helps!

If you are new to Prime numbers then watch the following video on  Basics of Prime Numbers

https://www.youtube.com/watch?v=WSx0QaGYv8M

If p and q are prime numbers, how many divisors does the pro

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