*Yesterday, Manhattan GMAT posted a GMAT question on our blog. Today, they have followed up with the answer:*

Let’s first figure out what pairs of integers “succeed” – that is, the product of the two integers is of the desired form

*a*2 –*b*2. As a matter of instinct, we should immediately factor*a*2 –*b*2 into (*a*+*b*)(*a*–*b*), which is a product. Thus, we need one of the two chosen integers (specifically, the larger one) to match*a*+*b*, while the other integer matches*a*–*b.*Let’s say that we pick 2 and 5. Then

*a*+*b*= 5 and*a*–*b*= 2. When we solve for*a*and*b*, we get*a*= 3.5 and*b*= 1.5. So the product of 2 and 5 (i.e., 10) cannot be written as the difference of two perfect squares,*a*2 –*b*2, and the pair {2, 5} would be an unsuccessful pair of chosen integers.Rather than calculate

*a*and*b*for every pair of integers, let’s consider how we might have solved for*a*in the previous case. We could have added the two equations, yielding 2*a*= 2 + 5 = 7. We can see right away that*a*cannot be an integer, because 7 is an odd number. In fact, whenever the sum of the two chosen integers is odd, we will not get integer values of*a*and*b*. On the other hand, when the sum of the two chosen integers is even, we will get integer values of*a*and*b*.We need to find out how many of the products can be expressed as either the product of two odd or two even factors. All the possible products are 10 (2×5), 14 (2×7), 16 (2×8), 35 (5×7), 40 (5×8), and 56 (7×8).

16 = 2 × 8.

35 = 5 × 7.

40 = 4 × 10.

56 = 4 × 14.

There are 6 possible products, and 4 of them can be expressed as the product (

16 = 2 × 8.

*a*= 5 and*b*= 335 = 5 × 7.

*a*= 6 and*b*= 140 = 4 × 10.

*a*= 7 and*b*= 356 = 4 × 14.

*a*= 9 and*b*= 5There are 6 possible products, and 4 of them can be expressed as the product (

*a*+*b*)(*a*–*b*).Thus, the probability we want is 4/6 or 2/3.

**The correct answer is A.**