The mean and the median of seven distinct positive integers is 30. If

Question

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

A 111
B 180
C 210
D 891
E 900

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Vishalp05 · Answer

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.

Using the same concept the numbers need to be around the mean
The mean and median of seven distinct positive integers is 30. Since the integers are consecutive, the data values are 27, 28, 29, 30, 31, 32, and 33. The product of the smallest and largest integers is 27 * 33 = 891.

niels.v_ · Answer

Important thing to note is that when the median and the mean are equal, it means the numbers in the set are equally spaced. When you combine this with the fact that the seven integers are such that their product is maximum, we know that the numbers are closely spaced. 27,28,29,30,31,32,33 is the correct set. You can always recalculate the average in case you are un sure about your decision which in this case is correctly 30. We multiple 27 by 33 to get 891. D

GMATinsight · Answer

- - - - - - -

Mean = 30
i.e. Sum = 7*30 = 210
  for maximum product when we have sum of those number, the numbers should be equally distributed
e.g. if a+b = 10
then a*b maximum = 5*5 = 25

Here the numbers are distinct so they should be evenly spread around the median

i.e. numbers should be {27, 28, 29, 30, 31, 32, 33}

Product of smallest and Maximum = 27*33 = 891

Answer: Option D

GMATinsight · Answer

Hi  Vishalp05

I think there is a typo error  (highlighted)

saynchalk · Answer

Small typo my friend, but perfect explaination Let me know when you've edited it, I'll remove my comment too

The mean and the median of seven distinct positive integers is 30. If

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The mean and the median of seven distinct positive integers is 30. If

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