**Case 1: Non square no:**

Let’s take a simple no: 84.

Factors of 84 are:

1,2,3,4,6,7,12,14,21,28,42,84.

Total: 12 Let’s check if we miss any factor:

Total no of factors for 84: [(2^2)*3*7] = 3*2*2=12.

So, we captured all the factors of 84.

Now 84 can be expressed as a product of 2 natural no’s:

1*84, 2*42, 3*28, 4*21, 6*14, 7*12 ------- 6 Ways So if a no has n factors, it can be expressed as a product of 2 natural no’s as:

---- {(First factor from left)*(first factor from right)} ,

{(second factor from left)*(second factor from right)} ,

till………………………. {([n/2]nd factor from left) * ([n/2]nd factor from right)}.

**So, if a no has ‘N’ factors, where N is even (for non square number), total no of ways it can be expressed as product of 2 natural no’s: N/2.**

If you are clear till this point, you are thinking about what if a no has odd no of factors.

This leads to our case no 2.

**Case 2: Square number:** As many of you might be aware that **Only **a square number has odd no of factors. Or if a no has odd no of factors, then it has to be a square no.

lets take a simple no: 36

Factors of 36: 1,2,3,4,6,9,12,18,36.

Total no of factors: 9.

36 can be expressed as a product of 2 different natural no’s:

1*36, 2*18, 3*12, 4*9, 6*6. ------ 5 ways.

If you notice, middle no is expressed as a product to itself (6*6).

**So, for a square no having no of factors as N, total no of ways are:**

** (N+1)/2 ------** **If repetition of numbers is allowed.**

** (N-1)/2-------** **If both the no’s has to be unique.**

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**Pratik Shah**(appeared for CAT 2016: DILR:99.86, Quant:95.9 percentile) Teaching Quant and DILR for CAT since April ,2017.

Contact on:

pratikquant@gmail.com

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