 # MSC #22: The Difference of Two Squares

“In how many ways can 96 be expressed as the difference of two square integers? A) 0;  B) 1;  C) 2; D) 3;  E) 4”

This question in the Intermediate Level of the 2nd International Vedic Mathematics Olympiad is just another example of the 22nd of our 30 MATH-Inic Specials for Christmas series.

Our Math Special for today is the other half of “Using the average”. This time we will start with the difference of two squares or DOTSa2 – b2 = (a + b) (a – b).

In his book, Vedic Mathematics, Sri Bharati Tirthaji wrote that given this form, any number can be readily expressed as a difference of two squares because it is very easy to express any given number as a product of two numbers. Even if a number is prime, it can be expressed as a product of itself and 1, as in 11 = 11 x 1.

(a + b)2 – (a – b)2 = (a2 + 2ab + b2) – (a2 – 2ab + b2) = 4ab, we can now express ab as equal to [(a + b)2 / 4] – [(a – b)2/4] or [(a + b)/2]2 – [(a – b)/2]2

If we want to express a number as a difference of two squares, we can list down its factor pairs. For example, 24 can be factored into 24 x 1, 12 x 2, 8 x 3 and 6 x 4. If we want to use only the squares of integers, we will choose the pairs which are both even or both odd numbers: 12 x 2 and 6 x 4, in this case.

For 12 x 2, we have (12 + 2)/2 = 7 and (12 – 2)/2 = 5; 72 – 52 = 24

For 6 x 4, we have (6 + 4)/2 = 5 and (6 – 4)/2 = 1; 52 – 12 = 24

If we used 24 x 1, we would get (25/2)2 – (23/2)2 = 12.52 – 11.52 . We can simplify the computation by using By One More Than the One Before: 12(13) – 11(12) = 2(12) = 24. (Note that 12.52 = 156.25 and 11.52 = 132.25; the decimal parts will cancel each other out).

On the other hand, if we used 8 x 3, we would have (11/2)2 – (5/2)2 = 5.52 – 2.52 = 5(6) – 2(3) = 30 – 6 = 24.

And since we can also express 24 as 48 x 1/ 2, 72 x 1/ 3, 96 x 1/ 4, etc. we can have an infinite number of ways to express a number as a difference of two squares.

For the IVMO 2022 question, we just need to find out how many factor pairs of 96 are both even or both odd. But since 96 is even, both factors must be even. Listing the factor pair of 96, we have:

96       = 1 x 96

= 2 x 48

= 3 x 32

= 4 x 24

= 6 x 16

= 8 x 12

So the answer is E) 4.