# Answer to Descriptive Statistics, Prime Numbers, Progression

The ages of three friends are prime numbers. The sum of the ages is less than 51. If the ages are in Arithmetic Progression (AP) and if at least one of the ages is greater than 10, what is the difference between the maximum possible median and minimum possible median of the ages of the three friends?

(A) 0
(B) 1
(C) 13
(D) 6
(E) 8

The ages are prime numbers in arithmetic progression i.e., they have a common difference. Furthermore, at least one of them is greater than 10.
Let the ages be a, b, and c such that a < b < c.
a + b + c < 51

Since a, b, and c are in AP, a + b + c = 3b
3b < 51 or b < 17

The largest prime number less than 17 is 13.

The maximum median is 13 (when the ages are 3, 13, and 23 OR 7, 13, and 19).
The minimum median is 7 (when the ages are 3, 7, and 11)

Remember that 3, 5, and 7 is not accepted as atleast one number has to be greater than 10.

The required difference is 13 – 7 = 6.

The corrcet answer is D