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# If a, b, c and n are positive integers and m=a4b3cn, how many factors

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If a, b, c and n are positive integers and m=a4b3cn, how many factors  [#permalink]

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23 Mar 2017, 00:15
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55% (hard)

Question Stats:

32% (00:44) correct 68% (01:01) wrong based on 37 sessions

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If a, b, c and n are positive integers and $$m=a^4*b^3*c^n$$, how many factors does m have?

1) a, b, and c are prime numbers

2) n=2

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Re: If a, b, c and n are positive integers and m=a4b3cn, how many factors  [#permalink]

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23 Mar 2017, 03:02
ziyuen wrote:
If a, b, c and n are positive integers and $$m=a^4*b^3*c^n$$, how many factors does m have?

1) a, b, and c are prime numbers

2) n=2

Hi

1) a, b, and c are prime numbers

There is no information about "n" and we are not directly told that they are distinct primes. Insufficient.

2) n=2

a, b, c can be primes or composite. Insufficient.

(1)&(2) We know that a, b and c are primes, we know the power of c, but we still don't know wheather a,b,c are distinct primes or not:

Say: $$2^4*b^3*5*2$$ -----> $$5*4*3 = 60$$ factors.

or $$2^4*3^3*3^2 = 2^4*3^5$$ -------> $$5*6 = 30$$ factors. Still insufficient.

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Re: If a, b, c and n are positive integers and m=a4b3cn, how many factors  [#permalink]

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23 Mar 2017, 14:32
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ziyuen wrote:
If a, b, c and n are positive integers and $$m=a^4*b^3*c^n$$, how many factors does m have?

1) a, b, and c are prime numbers

2) n=2

Target question: How many factors does m have?

Given: m = (a^4)(b^3)(c^n)

The two statements seem to provide a lot of information, so I'm going to jump straight to..

Statements 1 and 2 combined
There are several values of a, b, c, and n that satisfy BOTH statements. Here are two:
Case a: a = 2, b = 2, c = 2 and n = 2, in which case m = (2^4)(2^3)(2^2) = 2^9. In this case, m has 10 factors.
Case b: a = 3, b = 3, c = 2 and n = 2, in which case m = (3^4)(3^3)(2^2) = (3^7)(2^2). In this case, m has 24 factors.
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

---------ASIDE---------------
To determine the number of factors of m, I used the following rule:

If N = (p^a)(q^b)(r^c)..., where p, q, r,...(etc.) are prime numbers, then the total number of positive divisors of N is equal to (a+1)(b+1)(c+1)...

Example: 14000 = (2^4)(5^3)(7^1)
So, the number of positive divisors of 14000 = (4+1)(3+1)(1+1) = (5)(4)(2) = 40
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Re: If a, b, c and n are positive integers and m=a4b3cn, how many factors   [#permalink] 23 Mar 2017, 14:32
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