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

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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.
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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