Divisors - PS

[karthikpandian19]

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

(A) 9 (B) 12(C) 18(D) 28(E) 36[/list]

[Anurag@Gurome]

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

(A) 9 (B) 12(C) 18(D) 28(E) 36[/list]

When we have to find the number of divisors of a number, then factorize the number into its prime factors, like, n = a^x * b^y, where a, b are distinct prime factors of n, and x, y are powers of prime factors a and b respectively.
Then number of divisors of n = (x + 1)(y + 1)

Here number of divisors of p^3 * q^6 = (3 + 1)(6 + 1) = 4 * 7 = 28

The correct answer is D.

[karthikpandian19]

Anurag,

Can this be taken as like this also?
n = a^x * b^y implies number of divisors of n = (x + 1)(y + 1)

Is it, n = a^x implies number of divisors of n = (x + 1)

How does it work for a number 36?

36 = 2^2 * 3^2
Which implies there should be (2+1) (2+1)
ie. 9 divisors

Divisors of 36 - 2,3,4,6,9,12,18,36 (we have only 8, is "1" also to be done included in each instances.?????

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

(A) 9 (B) 12(C) 18(D) 28(E) 36[/list]

When we have to find the number of divisors of a number, then factorize the number into its prime factors, like, n = a^x * b^y, where a, b are distinct prime factors of n, and x, y are powers of prime factors a and b respectively.
Then number of divisors of n = (x + 1)(y + 1)

Here number of divisors of p^3 * q^6 = (3 + 1)(6 + 1) = 4 * 7 = 28

The correct answer is D.

[Anurag@Gurome]

Anurag,

Can this be taken as like this also?
n = a^x * b^y implies number of divisors of n = (x + 1)(y + 1)

Is it, n = a^x implies number of divisors of n = (x + 1)

How does it work for a number 36?

36 = 2^2 * 3^2
Which implies there should be (2+1) (2+1)
ie. 9 divisors

Divisors of 36 - 2,3,4,6,9,12,18,36 (we have only 8, is "1" also to be done included in each instances.?????

Yes, that's correct, when we count we have to include 1 and the number itself.

[gunjan1208]

Hi Anurag,

Can we take it as a rule everytime we see this kind of question?

Regards

[Anurag@Gurome]

Hi Anurag,

Can we take it as a rule everytime we see this kind of question?

Regards

Yes, definitely.

[GMATGuruNY]

This question should state that p and q are DISTINCT prime numbers. I posted an explanation here:

https://www.beatthegmat.com/divisors-t85731.html

[Elena89]

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

(A) 9 (B) 12(C) 18(D) 28(E) 36[/list]

The question is asking to find the total factors of the product P^3*q^6

since p and q are prime numbers, lets take p=2 and q=3

=> 2^3*3^6

total factors = (total possibilities for 2 to be in a factor of the product)*(total possibilities for 3 to be in a factor of the product)

=> total factors = (none + 3)*(none + 6)
= (1 + 3)*(1 + 6)
= 4*7
= 28

Hence the correct option: D) 28