Factors

[brick2009]

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??

[anandr84]

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

[heshamelaziry]

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

[heshamelaziry]

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.

[Ian Stewart]

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.