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Counting Quadruplets (Posted on 2010-08-20) Difficulty: 3 of 5
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.

No Solution Yet Submitted by K Sengupta    
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Solution solution Comment 11 of 11 |
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 B C D 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
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