A 5 -digit number is formed by using the digits 1, 2, 3, 4, 5 and an unknown digit x (where x is an integer and 5 < x 9 ) without repetition. If the total number of such 5 -digit numbers that are divisible by 6 is exactly 96 , then the value of x is

Step-by-step solution

The given digits are 1, 2, 3, 4, 5, x . The sum of the known digits is 1 + 2 + 3 + 4 + 5 = 15 . The total sum of all six digits is 15 + x . To form a 5 -digit number, one digit y must be excluded. The sum of the chosen 5 digits will be 15 + x - y . For the number to be divisible by 6 , it must be divisible by 2 (so its unit digit is even) and by 3 (so the sum of its digits is a multiple of 3 ). Thus, 15 + x - y 0 3 , which implies x y 3 . Let us test the possible values for x 6, 7, 8, 9 : If x = 9 , then y 0 3 . Th