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?

Answer 1

To maximize the product of the smallest and largest integers, the smallest integer should be 1 and the largest should be 209, resulting in a product of 209.

Step-by-step explanation:

To find the product of the smallest and largest integers, we need to determine the actual values of the integers. Since the mean and median are both 30, we know that the sum of the seven integers is 30 multiplied by 7, which is 210. We want the integers to be as spread out to maximize the product. Let's assume the smallest integer is m and the largest is n. So we have m + n = 210. To maximize the product, we need to maximize the difference between m and n, which means they should be as far apart as possible. If we choose m = 1 and n = 209, the product will be 1 × 209 = 209. Therefore, the product of the smallest and largest integers is 209.

Answer 2

891

Step-by-step explanation:

It is mentioned that the mean and median is 30 for seven distinct positive integers and we know that median is the middle value exists in the data and we can asses the data values as we know that the data value must be consecutive. So the data values will be 27,28,29,30,31,32,33

Also mean is sum of data values divided by the number of data values so mean=30

sum of value/n=30

sum of values/7=30

sum of values=30*7=210

and sum of assessed data values is 27+28+29+30+31+32+33=210. Hence the data values are verified.

Hence the product of smallest and largest integer=27*32=891