# CAT Practice : Number System: Factors

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Number has 15 factors, how many factors might square root of that number have. These open-ended questions are very instructive.

## Number Theory - Factors

Q.11: A number N2 has 15 factors. How many factors can N have?
1. 5 or 7 factors
2. 6 or 8 factors
3. 4 or 6 factors
4. 9 or 8 factors

Choice B. 6 or 8 factors.

## Detailed Solution

Any number of the form paqbrc will have (a + 1) (b + 1)(c + 1) factors, where p, q, r are prime. (This is a very important idea)
N2 has 15 factors.
Now, 15 can be written as 1 * 15 or 3 * 5.

If we take the underlying prime factorization of N2 to be paqb, then it should have (a + 1) (b + 1) factors. So, N can be of the form

p14 or p2q4
p14 will have (14 + 1) = 15 factors
p2q4 will have (2 + 1) * (4 + 1) = 15 factors.

Importantly, these are the only two possible prime factorizations that can result in a number having 15 factors.

Correct Answer: 6 or 8 factors.

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## More questions from Number System - Factors

Dealing with factors of a number is a vital component in this topic. A range of questions can be asked based on this simple concept.