This weekend, I was presented with the following math puzzle:
Humphrey has a phone number abc-def-ghij such that each digit is unique. Furthermore, it holds the property such that: a is divisible by 1, ab is divisible by 2, abc is divisible by 3, and so on, so that abcdefghij is divisible by 10. What is the number?
It took me a couple hours to figure out, as I systematically plowed through a means of avoiding the brute force approach. Rather than sharing the answer, though, I thought I'd give the following hints, which might seem obvious, but are easily glossed over as you focus on finding the answer.
  1. The digit for a and i don't matter since 1 divides all natural numbers and 1 + 2 + 3 + ... + 9 = 45 which is divisible by 9.
  2. A number is divisible by 3 if the sum of its digits add up to a number divisible by 3.
  3. A number is divisible by 6 if it is divisible by both 2 and 3.
  4. A number is divisible by 4 if the last two digits form a number divisible by 4.
  5. A number is divisible by 8 if the last three digits form a number divisible by 8.
When you use the above properties (and their implications), you can really narrow down from the initial 10! = 3,628,800 the possibilities.

Happy hunting.

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