Problem Solving

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) 36

Hi,
if a number N = a^p. b^q.c^r.... where a,b,c are prime factors, then the number of factors is given by:
(p+1)*(q+1)*(r+1)*...
Here the number is p^3*q^6
So, number of factors is (3+1)8(6+1) = 28

Hence, D

Frankenstein wrote:Hi,
if a number N = a^p. b^q.c^r.... where a,b,c are prime factors, then the number of factors is given by:
(p+1)*(q+1)*(r+1)*...
Here the number is p^3*q^6
So, number of factors is (3+1)8(6+1) = 28

Hence, D

Thanks man. Does this formula apply to finding the factors in general? Like when you do prime factorization you can use it then to find the factors?

MBA.Aspirant wrote: Thanks man. Does this formula apply to finding the factors in general? Like when you do prime factorization you can use it then to find the factors?

Yes. That is my purpose of writing the formula. I have mentioned about the prime factors thing as well.

MBA.Aspirant wrote:

Frankenstein wrote:Hi,
if a number N = a^p. b^q.c^r.... where a,b,c are prime factors, then the number of factors is given by:
(p+1)*(q+1)*(r+1)*...
Here the number is p^3*q^6
So, number of factors is (3+1)8(6+1) = 28

Hence, D

Thanks man. Does this formula apply to finding the factors in general? Like when you do prime factorization you can use it then to find the factors?

It might be helpful to understand the reasoning behind the formula offered by Frankenstein.

To determine the number of positive factors of an integer:

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

For example:
72 = 2³ * 3².
Adding 1 to each exponent and multiplying, we get (3+1)*(2+1) = 12 factors.

Here's why:
To determine how many factors can be created from 72 = 2³ * 3², we need to determine the number of choices we have of each prime factor and to count the number of ways these choices can be combined:

For 2, we can use 2�, 2¹, 2², or 2³, giving us 4 choices.
For 3, we can use 3�, 3¹, or 3², giving us 3 choices.

Multiplying the number of choices we have of each factor, we get 4*3 = 12 possible factors.

Regarding the following 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) 36

The question needs to say that p and q are distinct prime numbers and that we are to count the number of positive divisors. Assuming p and q are distinct, we add 1 each exponent and multiply:
(3+1)(6+1) = 4*7 = 28.

If p and q are not distinct, then p³q� = p³p� = p�, yielding only 9+1 = 10 factors.

Thnks GMATGuruNY..
Was wondering why plus one ..

@finites ... 'plus 1' because of 2^0 and 3^0 (in choices) in GMATGuruNY's explanation.

GMATGuruNY wrote:

MBA.Aspirant wrote:

Frankenstein wrote:Hi,
if a number N = a^p. b^q.c^r.... where a,b,c are prime factors, then the number of factors is given by:
(p+1)*(q+1)*(r+1)*...
Here the number is p^3*q^6
So, number of factors is (3+1)8(6+1) = 28

Hence, D

Thanks man. Does this formula apply to finding the factors in general? Like when you do prime factorization you can use it then to find the factors?

It might be helpful to understand the reasoning behind the formula offered by Frankenstein.

To determine the number of positive factors of an integer:

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

For example:
72 = 2³ * 3².
Adding 1 to each exponent and multiplying, we get (3+1)*(2+1) = 12 factors.

Here's why:
To determine how many factors can be created from 72 = 2³ * 3², we need to determine the number of choices we have of each prime factor and to count the number of ways these choices can be combined:

For 2, we can use 2�, 2¹, 2², or 2³, giving us 4 choices.
For 3, we can use 3�, 3¹, or 3², giving us 3 choices.

Multiplying the number of choices we have of each factor, we get 4*3 = 12 possible factors.

Regarding the following 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) 36

The question needs to say that p and q are distinct prime numbers and that we are to count the number of positive divisors. Assuming p and q are distinct, we add 1 each exponent and multiply:
(3+1)(6+1) = 4*7 = 28.

If p and q are not distinct, then p³q� = p³p� = p�, yielding only 9+1 = 10 factors.

Thanks GMATGuru for your help. The part about the non-distinct factors was very helpful. I just faced it with 64 where the factors are/is 2^6. the number of factors then is (6+1) not (3+1) * (3+1)