Each of A, B, C and D is a positive integer with the proviso that A ≤ B ≤ C ≤ D ≤ 20.

Determine the total number of quadruplets (A, B, C, D) such that A*B*C*D is divisible by 50.

In order for the product to be divisible by 50, there need be the factors 2, 5 and 5. These factors may be distributed with any other factors in any manner between the four positive integers A, B, C and D within the given provisio that A ¡Ü B ¡Ü C ¡Ü D ¡Ü 20.

In order for the factors 2, 5 and 5 to be represented, two of the numbers, A, B, C and D, must be equal to 5, 10, 15 or 20. If both of those two numbers are odd (i.e. 5 and 5, 5 and 15, or 15 and 15), at least one of the other two numbers must have a factor of 2, i.e., be even.

The total number of ordered combinations, i.e., quadruplets, is 1570.

Posted by Dej Mar on 2010-08-21 04:36:06