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?
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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