Minimum Integer

of

Dividing a Five Digit Integer

by the

Sum of its Digits

 

PROBLEM:

What is the smallest integer that can result from dividing a 5 digit integer by the sum of its digits.

Prove the minimum is 407 by some means other than running a spreadsheet or a computer program to test all 90,000 five digit numbers.

Hint:   The sum of the digits ranges from  1 to 45.   What is the smallest 5-digit number with the sum of its digits equal to  15?   35? 40? others?   What is the quotient (not necessarily an integer) when the smallest 5-digit number with the sum of its digits  =  k   is divided k?   The smallest integer quotient will have to be slightly larger.


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