Other Topics Puzzle – Challenge 100

Challenge:

What is the greatest four-digit integer that meets the following three restrictions?

1- All of the digits are different.

2- The greatest digit is the sum of the other three digits.

3- The product of the four digits is divisible by 10 and not equal to zero.

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First, notice rule 3 (The product of the four digits is divisible by 10) tells you that one digit must be 5 and none of them should be 0.
Since we’re looking for the greatest four digit integer we need to start with 9, which is the greatest digit. If the first digit is 9, then all the three other digits should add up to 9.
One digit is 5 and there is no 0. Therefore, there are two possible ways:
5 + 3 + 1 = 9
5 + 2 + 2 = 9
None of these solutions work. Because, in the first one, there is no even number and it’s not divisible by 10. In the second one, digit 2 repeated.
Therefore, 9 is not the first digit. Let’s try 8.
If 8 is the first digit. Then:
5 + 2 + 1 = 8
5 + 3 + 0 = 8
The second one has 0, so the only solution is the first one. The number has digits 8, 5, 2, and 1. The greatest such number is 8521