# Number System : How to find sum and number of factors of a Number

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**How to find sum and number of factors of a Number**

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You might have encountered with such questions in earlier CAT papers or any Test series. Well questions of such types requires prior knowledge of solving it. Otherwise you will just waste time trying to solve such questions and will end up with no results. So is there any formula for such questions ?. Infact there is a complete algorithm to solve such questions. If you follow algorithm and practice few questions, you will be able to solve it in within minutes.

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**Algorithm **

**Lets take a number 720. So the question is to “Find the sum and number of Factors of 720 .”**

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**Approach **

*1.Write the number in Standard Form.*

So , 720 = 2^4 x 3^2 x 5^1

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*2. Create number of brackets as the number of prime factors.*

*3. Fill each bracket with the sum of all the powers of the respective prime number starting from 0 to the highest power of that number.*

Sum of the factors = (2^0 + 2^1 + 2^2 + 2^3 + 2^4) x (3^0 + 3^1 + 3^2) x (5^0 + 5^1)

= (1 + 2 + 4+ 8 +16) x (1 + 3 + 9) x (1+ 5)

= 31 x 13 x 6 = 2418

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**How to find Number of Factors of a Number ?**

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If you will check the sum of factors in step 3, and expand the bracket, you will find that all are actually factors of the number.

Lets take a small number for this say 80

So, 80 = 2^4 x 5^1

Sum of the factors = (2^0 + 2^1 + 2^2 + 2^3 + 2^4) x (5^0 + 5^1) = 31 x6 = 186

= 2^0 x 5^0+ 2^1 x 5^0 + 2^2 x 5^0 + 2^3 x 5^0 + 2^4 x 5^0 + 2^0 x 5^1+ 2^1 x 5^1 + 2^2 x 5^1 + 2^3 x 5^1 + 2^4 x 5^1

= 1+ 2+4 +8 + 16 + 5 + 10 + 20 + 40 + 80 = 186

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So to calculate the number of factors of 80.

*4. Add 1 to the powers of prime factors and multiply it *

No. of Factors of 80 = (4+1) x (1+1) = 10.

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**Lets get back to the original question ” Find number of Factors of 720 ” **

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So , 720 = 2^4 x 3^2 x 5^1

Number of Factors of 720 = (4+1) x(2+1) x(1+1) = 5 x 3 x 2 = 30.

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**Number System : How to find sum and number of even and odd factors of a Number**

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