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Find All Numbers (Posted on 2006-07-25) Difficulty: 3 of 5
Find all four–digit numbers in which the product of the digits is equal to the sum of the digits, and the number itself is divisible by the sum of its digits.

No Solution Yet Submitted by Ravi Raja    
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Solution Solution | Comment 2 of 7 |

4112 and -4112.

As the product is equal to the sum, the digits must be 1,1,2,4. No other set of digits could offer a solution.  The following is the list of positive numbers where the product  and sum of the digits are equal: 1124, 1142, 1214, 1241, 1412, 1421, 2114, 2141, 2411, 4112, 4121, & 4211.  Of these the only 4112 is evenly divisible by 8.  As the problem does not require the number (only the digits) to be positive, -4112 is also valid as it is also a four-digit number divisible by the sum of its digits. 

  Posted by Dej Mar on 2006-07-25 11:52:10
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