How many different factors does the integer n have?

ohhh I didnt read the question correctly...anyways thanks a lot...

Thanks for clarification, I've missed \(a^0\) and \(b^0\)

Nice Question.
Here is what i did in this one.
We need to get the number of factors of positive integer n.

Statement 1=>
As a and b are "different" prime numbers => Number of factors of a must be 5*4=20
Hence sufficient .
Statement 2=>
There exist ∞ numbers with the same set of prime numbers.
E.g
5*7=> Four factors.
5^2*7^2=> Nine factors.
Etc.
Hence not sufficient.

Hence A.

(1) says that n = (a^4)(b^3), where a and b are different positive prime numbers.

So, with this, clearly we can find the actual value of n, and hence, the actual factors of n.

Question: How many different positive factors does the integer n have?

Number of factors of a number \(= (p+1)*(q+1)*(r+1)...\)
if, Number \(= a^p*b^q*c^r\)... where a, b, c,... are distinct prine factors of number n

Statement 1:\(n = a^4b^3\), where a and b are different positive prime numbers.

i.e. Number of factors of \(n = (4+1)*(3+1) = 12\)

SUFFICIENT

Statement 2:The only positive prime numbers that are factors of n are 5 and 7
But the exponents of 5 and 7 are unknown hence

NOT SUFFICIENT

Answer: Option A

Solution

Given:

• The number n is an integer

To find:

• The number of different positive factors of n

Analysing Statement 1
From the information given in statement 1, \(n = a^4b^3\), where a and b are different positive prime numbers.

• Therefore, number of factors of n = (4 + 1) x (3 + 1) = 5 x 4 = 20

Hence, statement 1 is sufficient

Analysing Statement 2
From the information given in statement 2, the only positive prime factors of n are 5 and 7.

• However, we don’t know the powers of each of those prime factors.

Hence, statement 2 is not sufficient.

The correct answer is option A.

Answer: A

Thank you, Bunuel for the clarifications. Just one question: The formula to find the number of factors returns only to positive factors, right? If negative prime factors would be allowed, then we would need to take them into account as well, wouldn't we? Is there a formula to find both positive and negative factors at the same time?
Thanks!

Most of the time we consider positive prime factorization for positive numbers that means you don't have to consider negative prime factors unless question demand something else.
For ex a^2*b^3 is the prime factorization of a number X
We know X is a positive no. So A^2 and B^3 have to be of same sign either negative or positive. But A^2 will be positive always as squaring a no.makes it positive. So B^3 has to be positive.
Therefore,
The prime factorization of X = (+)A^2*(+)B^3
If B^3 is positive B has to be positive.
But if a^2 is positive a can be both negative and positive, depending upon what's been asked.
So we can say the number X can only have negative prime number if its power is in even. (±a)^2 = a^2
There is no separate requirement for determining negative factors. You just have to look into the power of the factors. You will be able to determine what can be the signs of factors.

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I am confused with the question.
Is the question asking us to calculate the number of different positive factors (which can be calculated by formula) or calculate different factors (prime factors and non-prime factors)?

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