Q. 3 B5.0( 1 Vote )
Some natural numbers can be written as a difference of two perfect squares in two ways. For example.
24 = 72 – 52 = 52 – 12
32 = 92 – 72 = 62 – 22
40 = 112 – 92 = 72 – 32
In how many different ways can we write multiples of 16, starting with 48 as the difference of two perfect squares?
Answer :
Let’s use algebra. starting with x,y, the square of the difference is 
= 
+ 
- 2
the square of the sum is
= 
+ 
+ 2
what if we subtract the square of the difference from the square of the sum.
- 
= (
+ 
+ 2
- (
+ 
- 2
)
- 
=
writing this in reverse
= 
- 
multiples of 16 are 16,32,48,64,80…..starting with 48
48 = 4
12
1
= here x=12 and y=1
= 
- 
= 
- 
= 
- 
similarly 48 can also be written as
48 = 4
4
3
= here x=4 and y=3
= 
- 
= 
- 
= 
- 
48 can also be written as
48 = 4
6
2
= here x=6 and y=2
= 
- 
= 
- 
= 
- 
Similarly for 64 = 4
16
1 = 
- 
64 = 4
8
2 = 
- 
64 = 4
4
4 = 
- 
Similarly we can do it for other multiples of 16.
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