Data Sufficiency

How many different factors does the integer n have ?

1. n=a^4b^3 , where a and b are different positive prime numbers.
2. The only positive prime numbers that are factors of n are 5 and 7 .

Hi Madhur,
Whenever u post a question , pls post the OA within the spoilers
That way we dont have to wait and can move on.


thanks
-V

I got A.

here is my soln.

Stmt 1 says
n = a^4 *b^3 where a and b are diff prime numbers.

So total number of factors = (4+1)*(3+1) = 20.

sufficient.


Stmt 2:
The only positive prime numbers that are factors of n are 5 and 7 .


So we can write it as
n = 5^a * 7^b.
Since we dont know a and b, we cannot find out the total # of factors.

So not sufficient.

Hence A

I got A.

here is my soln.

Stmt 1 says
n = a^4 *b^3 where a and b are diff prime numbers.

So total number of factors = (4+1)*(3+1) = 20.

sufficient.


Stmt 2:
The only positive prime numbers that are factors of n are 5 and 7 .


So we can write it as
n = 5^a * 7^b.
Since we dont know a and b, we cannot find out the total # of factors.

So not sufficient.

Hence A

Vittal,

Is there a standard formula for number of factors. You used (4+1)*(3+1) = 20. So is it simply (power of X+1)*(power of Y+1)=# of factors?

Also, with B it's saying that there are only 2 prime factors 5 and 7. To me, that means that there are no other factors since every other number can be simplified to a prime factor (ex, 6, 4, 8, 9) which is why I chose B. Is this correct?

Just wanted to point statement one actually says (1. n=a^4b^3 , where a and b are different positive prime numbers.) There should be a * between the powers because it reads like you are raising a to the 4 raised to b to the 3.

Yes, there is a standard formula.
If you decompose a number N to its prime factors such that
N = a^i * b^j * c*k..., where a, b and c are prime numbers.
and i, j , k are +ve integers, then the total number of factors is
(i+1) * (j+1) *(k+1).

U are already wondering why we add 1 to the powers. This +1 accounts for 0th power of the corr prime number. ie a^0 =1 is also a factor of N.



in a^4b^3 in this problem the * is implicit.. Pls read it as
a^4 *b^3

Also, with B it's saying that there are only 2 prime factors 5 and 7. To me, that means that there are no other factors since every other number can be simplified to a prime factor (ex, 6, 4, 8, 9) which is why I chose B. Is this correct?

From B we only know the prime factors. From that we cannot predict all the factors. ie we need to know the powers of these prime factors to find the number of prime factors.
Put this another way: From B, we know that 5 and 7 are prime factors.
But we dont know if N = 5*7 or N=5^a * 7^b.
So this is insufficient.

HT Helps



Ht Helps

Thanks Vittal. That helps.

Is the question stem not asking "how many different factors are there?" ...so both 1 & 2 are sufficient, right?

GID09

GID09 wrote:Is the question stem not asking "how many different factors are there?" ...so both 1 & 2 are sufficient, right?

GID09

The question is asking "how many different factors are there"
NOT
"how many different PRIME factors are there"
So 2 is insufficient.