Difference of Squares

MBA.Aspirant wrote:In how many ways can 8192 be written as a difference of two perfect squares?

If 8192 is expressed as a difference of two perfect squares a² and b², then,

• 8192 = (a² - b²) = (a - b)(a + b)

As 8192 is an even integer, atleast one of (a - b) and (a + b) has to be even.

Now, 8192 = 2^13
Hence, 8192 can be factored as any one of the following seven ways,

• 2^1 x 2^12
  2^2 x 2^11
  2^3 x 2^10
  2^4 x 2^9
  2^5 x 2^8
  2^6 x 2^7
  1 x 8192

Out of these seven factorizations, the last one cannot be expressed as (a - b)(a + b) but others can be.

Hence, 8192 can be written as a difference of two perfect squares in 6 different ways.

The correct answer is A.

Note 1: 1 x 8192 cannot be expressed as (a - b)(a + b) because if we do so we'll get non-integer values for a and b.

Note 2: Here is an example how others can result a way to express 8192 as a difference of two perfect squares. For example take the factorization, 8192 = 2^3 x 2^10 = 8*1024 = (a - b)(a + b)
Hence, (a - b) = 8 and (a + b) = 1024
Hence, a = 516 and b = 508
Thus, 8192 = (516)² - (508)²