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# A number has three distinct digits. The sum of the digits equals the product of the digits. How many three-digit numbers satisfy this condit

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881
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A number has three distinct digits. The sum of the digits equals the product of the digits. How many three-digit numbers satisfy this condition?

Dabae  Jul 7, 2015

#3
+26399
+15

Here's a proof that only the three digits 1, 2 and 3 are involved:

.

Alan  Jul 11, 2015
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#1
+80935
+10

I believe this might be all the permutations of the divisors of the perfect number, "6"

Note  1 + 2 + 3  = 6  =  1* 2 *3       so...the three digit numbers are

123, 132, 213, 231 312, 321

CPhill  Jul 7, 2015
#2
+91435
+5

Thanks Chris,

Mmm I can't think of any others either but I would like to see a proof or a concrete demonstration that there are no others.

Melody  Jul 11, 2015
#3
+26399
+15

Here's a proof that only the three digits 1, 2 and 3 are involved:

.

Alan  Jul 11, 2015
#4
+91435
+5

Thanks Alan, that is great

Melody  Jul 11, 2015

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