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Question 723502: i am a four digit number. the sum of my digits is same as the product of my digits. who am i?
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! Zeros are not allowed as digits because they would make the product of the digits zero, and if it is a 4-digit number, the sum of the digits cannot be zero.
It cannot have 4 odd digits because that would make for an odd product but an even sum.
It cannot have exactly 1 or exactly 3 odd digits because that would make the sum odd, while the even digit(s) would make the product even.
Having no odd digits does not work either.
The smallest product that can be made with no odd digits is
 which is larger that the corresponding sum,  ,
and replacing a different even, non-zero number, skews the balance even further in favor of the product.
We are left with 2 odd digits plus 2 even digits as the only option.
The product will be a multiple of 4.
If the 2 even digits are 2, their sum would also be a multiple of 4, and the sum of the odd digits would have to be a multiple of 4 too.
For that case, using 2, 2, 1, and 3 as the digits gives too large a product:
 while  ,
and making the odd numbers larger only makes it worse.
Using 4 and 2 as the even digits gives {4, 2, 1, 1} as a set of digit that work:
 and 
Increasing any odd digits, or any even digit, will make the product too large, so we are left to make permutations of the set of digits {4, 2, 1, 1}.
NOTE: This looks like a problem for the middle school level forum at artofproblemsolving.com.
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