Question

'A' is the set of natural numbers less than 100. 'B' is the subset of 'A' such that 'B' contains two digit numbers from set 'A' which are divisible by exactly two numbers. 'C' is the subset of 'B' such that 'C' contains numbers whose sum of digits is divisible by 2. 'D' is the subset of 'C' such that 'D' contains numbers whose sum of digits is a perfect square or a perfect cube. Then find the median of the set 'D'?

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Answer 1

A be the set of natural numbers less than 100.
$\therefore$ A = {1, 2, 3, 4, ... , 99}
Now, B is the subset of A such that B contains two digit numbers which are divisible by exactly two numbers from the set A .
$\because$ Only prime numbers are the numbers which are divisible by 1 and itself only.
$\therefore$ B = {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97}
Now, C is the subset of B such that C contains numbers whose sum of digits is divisible by 2.
$\therefore$ C = {11, 13, 17, 19, 31, 37, 53, 59, 71, 73, 79, 97}
Now, D is the subset of C such that D contains numbers whose sum of the digits is a perfect square or a perfect cube.
$\therefore$ D = {13, 17, 31, 53, 71, 79, 97}
$\therefore$ Median of the set D = 53