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

SOLUTION

Finding the Number of Factors of an Integer

First make prime factorization of an integer \(n=a^p*b^q*c^r\), where \(a\), \(b\), and \(c\) are prime factors of \(n\) and \(p\), \(q\), and \(r\) are their powers.

The number of factors of \(n\) will be expressed by the formula \((p+1)(q+1)(r+1)\). NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: \(450=2^1*3^2*5^2\)

Total number of factors of 450 including 1 and 450 itself is \((1+1)*(2+1)*(2+1)=2*3*3=18\) factors.
For more on number properties check: math-number-theory-88376.html

BACK TO THE ORIGINAL QUESTION:

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

(1) n is the product of two different prime numbers --> n=ab, where a and b are primes, so # of factors is (1+1)(1+1)=4. Sufficient.

(2) n and 2^3 are each divisible by the same number of positive integers --> 2^3 has 4 different positive factors (1, 2, 4, and 8) so n has also 4. Sufficient.

Answer: D.
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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.

Well, first of all \(a^1*b=a*b^1=ab\).

As for the powers: ask yourself can we says that 12=2^2*3 is the product of two different prime numbers? No.

n is the product of two different prime numbers means n = (prime 1)*(prime 2).

Hope it's clear.
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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.
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(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.
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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
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