# CAT Practice : Number System: Factors

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How many factors of number N are also multiples of number K? Have you ever thought of that? Well, now you should.

## Number Systems - Factors

Q.5: How many factors of the number 28 * 36 * 54 * 105 are multiples of 120?
1. 540
2. 660
3. 594
4. 792

Choice C. 594.

## Detailed Solution

The prime factorization of 28 * 36 * 54 * 105 is 213 * 36 * 59.

For any of these factors questions, start with the prime factorization. Remember that the formulae for number of factors, sum of factors, are all linked to prime factorization.

120 can be prime-factorized as 23 * 3 * 5.

All factors of 213 * 36 * 59 that can be written as multiples of 120 will be of the form 23 * 3 * 5 * K.

213 * 36 * 59 = 23 * 3 * 5 * K
=> K = 210 * 35 * 58.

The number of factors of N that are multiples of 120 is identical to the number of factors of K.

Number of factors of K = (10 + 1) (5 + 1) * (8 + 1) = 11 * 6 * 9 = 594

## Alternative approach

Any factor of 213 * 36 * 59 will be of the form 2p * 3q * 5r. When we are trying to find the number of factors without any constraints, we see that.

p can take values 0, 1, 2, 3......13 – 14 values.
q can take values 0, 1, 2......6 – 7 values.
r can take values 0, 1, 2, 3......9 – 10 values.

So, the total number of factors will be 14 * 7 * 10.

This is just a rehash of our formula (a + 1) (b + 1) (c + 1).

In this scenario we are looking for factors of 213 * 36 * 59 that are multiples of 120. These will also have to be of the form 2p * 3q * 5r. But as 120 is 23 * 3 * 5, the set of values p, q, r can take are limited.

p can take values 3, .......13 – 11 values
q can take values 1, 2, .......6 – 6 values
r can take values 1, 2, 3, .......9 – 9 values

So, the total number of factors that are multiples of 120 will be 11 * 6 * 9 = 594.

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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.