In how many ways can 9898989898 be described as the difference of the squares of 2 natural numbers
Answers
Well, it is always true that the difference between squares of an odd no. as nd an even no. is also an odd no. and the difference between squares of two odd nos. or two even nos. is a multiple of 4.
Consider two odd nos. 2a + 1 and 2b + 1.
(2a + 1)² - (2b + 1)²
= 4a² + 4a + 1 - 4b² - 4b - 1
= 4(a² + a - b² - b)
Thus the difference is a multiple of 4.
Consider two even nos. 2a and 2b.
(2a)² - (2b)²
= 4(a² - b²)
This the difference is a multiple of 4.
Consider an odd no. 2a + 1 and an even no. 2b.
(2a + 1)² - (2b)²
= 4a² + 4a + 1 - 4b²
= 4(a² - b² + a) + 1
Thus the difference is an odd no. and this odd no. leaves remainder 1 on division by 4.
From this we can say that, those who leave remainders 2 and 3 on division by 4 can't be written as the difference of two perfect squares.
Here the number 9898989898 leaves remainder 2 on division by 4. So it can't be written as the difference of two perfect squares.