# CAT Practice : Number System: Factors

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How many even factors does a number have? Now thats a rather odd question. Think about that.

## Number Theory - Even Factors

Q.6: Number N = 26 * 55 * 76 * 107; how many factors of N are even numbers?
1. 1183
2. 1200
3. 1050
4. 840

Choice A. 1183.

## Detailed Solution

The prime-factorization of 26 * 55 * 76 * 107 is 213 * 512 * 76

The total number of factors of N = 14 * 13 * 7

We need to find the total number of even factors. For this, let us find the total number of odd factors and then subtract this from the total number of factors. Any odd factor will have to be a combination of powers of only 5 and 7.

Total number of odd factors of 213 * 512 * 76 = (12 + 1) * (6 + 1) = 13 * 7

Total number of factors = (13 + 1) * (12 + 1) * (6 + 1)

Total number of even factors = 14 * 13 * 7 - 13 * 7

Number of even factors = 13 * 13 * 7 = 1183

## Alternative approach

Any factor of 213 * 512 * 76 will be of the form 2p * 5q * 7r.

Any even factor of 213 * 512 * 76 will also be of the same form, except that p cannot be zero in this case.

p can take values 1, 2, 3, .........13 – 13 values
q can take values 0, 1, 2, .........12 – 13 values
r can take values 0, 1, 2, 3,….5 – 7 values

So, the total number of even factors be 13 * 13 * 7 = 1183

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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.