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.