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

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If p and q are prime numbers, how many divisors does the pro [#permalink]  01 Jul 2010, 15:48
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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
[Reveal] Spoiler: OA
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Re: number of divisors for prime numbers [#permalink]  01 Jul 2010, 16:06
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ksharma12 wrote:
If p and q are prime numbers, how many divisors p^3*q^6 does the product have?

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

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, $$p^3*q^6$$ will have $$(3+1)(6+1)=28$$ different positive factors.

Hope it helps.
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Re: number of divisors for prime numbers [#permalink]  22 Sep 2013, 19:17
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Re: If p and q are prime numbers, how many divisors does the pro [#permalink]  13 Nov 2014, 15:08
Hello from the GMAT Club BumpBot!

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Re: If p and q are prime numbers, how many divisors does the pro [#permalink]  13 Nov 2014, 16:25
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
Re: If p and q are prime numbers, how many divisors does the pro   [#permalink] 13 Nov 2014, 16:25
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