Last Updated on May 11, 2023
GMAT OFFICIAL GUIDE PS
Solution:
Among all the pairs of integers that satisfy m^2 + p^2 < 100, the product mp is greatest if m and p are as close to each other as possible, and optimally, if they are equal. Thus, we can let p = m and our inequality becomes m^2 + m^2 < 100. Let’s solve it:
2m^2 < 100
m^2 < 50
m < √50
Since m is a positive integer and the largest positive integer less than √50 is 7, m = 7. In that case, p is also 7. Thus, the greatest possible value of mp is 7 x 7 = 49.
Alternate Solution:
Let’s test each answer choice, starting from the greatest, which is 51.
Notice that 51 = 3 x 17, so our only choices for m and p are 3 and 17 or 1 and 51. Neither of these choices satisfies m^2 + p^2 < 100, and therefore mp cannot equal 51.
Next, let’s test 49. Since the choice m = 49 and p = 1 does not satisfy m^2 + p^2 < 100, let’s take a look at m = 7 and p = 7. Since m^2 + p^2 = 49 + 49 = 98 < 100, mp can equal 49. Since we are looking for the greatest possible value of mp, 49 is correct.
Answer: D
