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Factors

by brick2009 » Tue Nov 10, 2009 8:28 pm
Q11: How many different factors does the integer n have?
(1) n = a4b3, where a and b are different positive prime numbers.
(2) The only positive prime numbers that are factors of n are 5 and 7.

OA: A

How??
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by anandr84 » Tue Nov 10, 2009 10:48 pm
1> n=a^4 . b^3
where a and b are different primes. So the factors of n will be
1, a, a^2,a^3, a^4 , b, b^2, b^3 and n itself
Thus sufficient

2> only positive primes tht are factors are 5 and 7. It doesnt mention about non prime factors. INSUFFICIENT


So A
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by heshamelaziry » Tue Nov 10, 2009 11:33 pm
anandr84 wrote:1> n=a^4 . b^3
where a and b are different primes. So the factors of n will be
1, a, a^2,a^3, a^4 , b, b^2, b^3 and n itself
Thus sufficient

2> only positive primes tht are factors are 5 and 7. It doesnt mention about non prime factors. INSUFFICIENT


So A

Could you explain this further ?

1> n=a^4 . b^3
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by heshamelaziry » Tue Nov 10, 2009 11:37 pm
anandr84 wrote:1> n=a^4 . b^3
where a and b are different primes. So the factors of n will be
1, a, a^2,a^3, a^4 , b, b^2, b^3 and n itself
Thus sufficient

2> only positive primes tht are factors are 5 and 7. It doesnt mention about non prime factors. INSUFFICIENT


So A

I see what you did with A, but you changed the statement, unless you know it was posted wrong. Also, 1 is not a prime number.
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by Ian Stewart » Fri Nov 13, 2009 1:19 pm
anandr84 wrote:1> n=a^4 . b^3
where a and b are different primes. So the factors of n will be
1, a, a^2,a^3, a^4 , b, b^2, b^3 and n itself
Thus sufficient

2> only positive primes tht are factors are 5 and 7. It doesnt mention about non prime factors. INSUFFICIENT

So A
There are quite a few other factors besides those listed above. If a and b are different primes, then the number (a^4)(b^3) will have 20 different positive divisors - it makes no difference what primes a and b are. The divisors will be:

1, a, a^2, a^3, a^4
b, ab, (a^2)b, (a^3)b, (a^4)b
b^2, a(b^2), (a^2)(b^2), (a^3)(b^2), (a^4)(b^2)
b^3, a(b^3), (a^2)(b^3), (a^3)(b^3), (a^4)(b^3)

So, for example, the number 432 = (2^4)(3^3) has 20 positive divisors, as does the number 648 = (2^3)(3^4).

In general, if you have a prime factorization, to count divisors you need only look at the exponents: add one to each and multiply, and you'll find how many divisors a number has. So if a prime factorization is in the form (p^4)(q^3), where p and q are different primes, then adding one to each exponent and multiplying we find that (p^4)(q^3) has 5*4 = 20 positive divisors.
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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