Solutions: sum of some numbers
June 3, 2007 pm30 7:15 pm
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There is a 5-digit number which is the sum of all 3 digit numbers made from its digits. What is it?
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JD2718 
I did some basic calculations to get a feel for what number it would be. In this case, probably a number with five distinct digits and no zeroes. Now, each digit is found 12 times in the first, second and third positions of the 3-digit numbers. So the sum of the 3-digit numbers will be (1200 + 120 + 12) multiplied by the sum of the digits. So I grabbed a calculator and put y = 1332x in a table until I found a value whose digits added up to x.
35964 is that number.
This can be accelerated a little bit. Note that y is divisible by 9, since 1332 is divisible by 9. If a number is divisible by 9, then the sum of its digits is also divisible by 9. Therefore, x is divisible by 9, and so y is a multiple of 1332*9
I got the same thing, but the same method — except that being old fashioned I used a FORTRAN routine instead of a calculator…
I didn’t notice the “sum of the digits is a factor of 9” until I looked at all the trials…
— Rachel