If n is an integer, then n is divisible by how many positive integers?

Statement 1: If n is a product of two prime number, then it must be divisible by 4 positive integers. Sufficient.

Statement 2: 2^3 is divisible by 4 positive integers and since n and 2^3 are divisible by same number of positive integers, this statement is sufficient as well.

Hence the answer is D.

Hi Bunuel, In statement-1, how do we know that the powers of two prime numbers is 1? I interpreted st-1 as n is a product of two different primes but their powers could be anything. So n could be ab, a^1*b, or a*b^1. Not Sufficient.

Haha! Off course. Thats why silly errors are killing me.



Yes makes sense. thanks!

What if statement 1: it is product of two prime number. Then can it be -2 and 5 then n= -10. Hence Positive factors would be 1, 5 and 10 (3 in all) whereas if it is 2 and 5 then it is, 1,2,5 and 10 (4 in all). No where they have mentioned that N has to be positive or product of two prime numbers has to be positive.

Looking for response. Bunuel @targettestprep

-10 is divisible by 2 too. Also, only positive numbers can be primes, so -2 is not a prime.

(1) n=ab (a and b are the prime numbers). Then n will be divisible by 1 and itself (like every integer) and by a and b. So we know it is divisible by four positive integers. (1) is sufficient.

Example with numbers: n=2*3=6. 6 is divisible by 1,2,3 and 6.

Note that we know that the only possible answers are A and D at this point. Even if we have no idea about (2) we already have a 50% chance to be correct.

(2) 2^3=8 By simply listing the divisors of 8 we will find out that they are 1, 2, 4 and 8.

So (2) is also sufficient.

The answer is D.

(2) n and 2^3 are each divisible by the same number of positive integers.
What if n is 36 ?
36 is divisible by the same number of positive integers as 2^3 but it has more positive integers

The number of factors of 36 = 2^2*3^2 is (2 + 1)(2 + 1) = 9, while the number of factors of 2^3 is 4. Not sure what you mean there.

This is a question on finding the number of factors of an integer.

It is therefore useful to know that Primes have exactly 2 factors, the number 1 has exactly one factor and, for a composite number N which can be expressed as N = \(a^p * b^q* c^r…..\), the number of factors = (p+1) (q+1) (r+1)….

Statement I alone says that n is the product of two different prime numbers.

It’s very important to not interpret this as “n has exactly two prime factors”. This is what causes a lot of people to think that the first statement is insufficient, which is incorrect.

Statement I is clear and categorical in saying that n is a product of two different prime numbers. Therefore, n is a composite number which can be written as n = \(a^1 * b^1\) where a and b are the two different prime numbers.

Number of factors of n = (1+1) * (1+1) = 4.

Statement I alone is sufficient to find the number of factors of n. Answer options B, C and E can be eliminated. Possible answer options are A or D.

Statement II alone says that each of n and \(2^3\) have the same number of factors.
\(2^3\) has a total of 4 factors, so n also has 4 factors.
Statement II alone is sufficient to find the number of factors of n. Answer option A can be eliminated.

The correct answer option is D.

Hope that helps!
Aravind B T

­But what if the product of two different prime numbers IS a prime number like 1*17? Thnx 

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A prime number cannot be written as the product of two prime numbers by definition, because a prime does not have a factor other than 1 and itself. Also, 1 is not a prime number. The smallest prime is 2.