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**How many ways a number can be expressed as a difference of two perfect squares of a natural no: **

** 1. Odd number**

** 2. Even no divided by 4**

** 3. Even no not divided by 4.**

**Odd No:**

take any odd no: for eg: 405

to express 405 as a difference of two perfect squares:

405= X2 –Y2 = (X-Y)(X+Y)

Replace X-Y=a and X+Y=b

so we need to find out in how many ways 405 can be expressed as a*b.

If you don’t know this concept, please read article:

https://www.pagalguy.com/discussions/hh-asd-fdf-fd-4771466512957440 No of Factors of 405 are: (34 * 51) = 5*2= 10

Factors(1,3,5,9,15,27,45,81,135,405).

So, no ways in which 405 can be expressed as product of 2 natural no’s= 10/2= 5.

(1*405,3*135,5*81,9*45,15*27)

So, no of ways in which 405 can be expressed as a difference of two perfect squares=5.

Let’s check: 405= 1*405.

X-Y=1

X+Y=405

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2X=406 ∴X=203 and Y=202.

∴ 405 can be expressed as a difference of two perfect squares:

2032 – 2022,

[(135+3)/2]2 - [(135-3)/2]2 = 692 -662

[(81+5)/2]2 - [(81-5)/2]2 = 432 -382.

[(45+9)/2]2 - [(45-9)/2]2 = 272 -182.

[(27+15)/2]2 - [(27-15)/2]2 = 212 -62.

From this we are clear that sum of 2 factors should be divided by 2. Hence both the factors should either be odd or even .**As for odd no, all the factors are odd, an odd number can be expressed as a difference of two perfect squares= no of ways in which a number can be expressed as product of 2 natural no’s= N/2.**

** Where N=number of factors.**

**Case2:**

**Lets take even number now which is divisible by 4:**

120= 1*120, 2*60, 3*40,4*30,5*24,6*20,8*15,10*12.

lets take a case 1*120.

1+120/2=121\2=60.5 and 120-1/2=59.5 .. Both of these are not natural no’s.

Hence we need to discard the cases where one factor is odd and one is even.

So only valid cases are: 2*60, 4*30 ,6*20, 10*12.

But, discard the cases where one factor is odd and one is even cannot be done manually each time.

Shortcut: as the no is even, at least one of the two factors will be even every time.

∴ we need to consider cases where both the factors are even . Hence express 120 as 2a’*2b’.

120=2a’ * 2b’

30=a’*b’

So no of ways to express 30 as product as two natural no’s = No of factors of 30/2 = 8/2 = 4 = no of ways in which 120 can be expressed as a difference of two perfect squares.

**For a number which is divisible by 4:**

** a. Divide the no by 4.**

** b. Find the number of ways in which N/4 can be expressed as product of 2 natural nos. **

**Case 3:**

**Now consider a case where even number is not divisible by 4.**

E.g.: take no as 50.

To find the answer, we need to express 50 as 2a’*2b’

Hence 50/4 = a’*b’. As 50 cannot be divided by 4, there won’t be any case in which 50 can be expressed as a product of 2 even natural nos.

**So, an even no which is cannot be divided by 4: cannot be expressed as a difference of squares of 2 natural nos.**

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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.

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