GMAT problem solving

Question

The mean and median of seven distinct positive integers is 30. If the seven intergers are such that their product is maximum, what is the product of the smallest and the largest integers?

Answer 1

For the product to be maximum for a set of numbers, they need to be close to each other. Having a big range (difference between the largest and the smallest number) would mean that the product will be not be maximum. Take an example of 3 numbers with a mean of 5. The numbers could be

4,5,6 product = 120

3,5,7 product = 105

2,5,8 product = 80

1,5,9 product = 45

0,5,10 product = 0

So as the range increases, the product decreases. Using the same concept the numbers need to be around the mean

which makes the 7 distinct integers as 27,28,29,30,31,32,33

so the answer is 27*33 = 891

please note a*b > (a-x)*(b+x)