Problem 24 · 2011 Math Kangaroo
Stretch
Number Theory
divisibilitycasework
Numbers are to be built using only the digits 1, 2, 3, 4 and 5 in such a way that each digit is only used once in each number. How many of these numbers will have the following property: the first digit is divisible by 1, the first 2 digits make a number divisible by 2, the first 3 digits make a number divisible by 3, the first 4 digits make a number divisible by 4, and all 5 digits make a number divisible by 5?
Show answer
Answer: A — It's not possible
Show hints
Hint 1 of 2
Each prefix has a divisibility rule: the 5th digit forces a 5 at the end, the even positions force even digits.
Still stuck? Show hint 2 →
Hint 2 of 2
Check whether the first-four-digits-divisible-by-4 rule can still be met.
Show solution
Approach: apply the prefix rules and hit a contradiction
- The 5-digit number must end in 5 (divisible by 5 with no 0 available), and digits 2 and 4 must be even, so they are 2 and 4.
- Divisibility of the first three digits by 3 forces digit 2 = 2, hence digit 4 = 4 and digits 1,3 are 1 and 3.
- Then the first four digits end in '14' or '34', and neither is divisible by 4 — a contradiction.
- So no such number exists: it's not possible.
Mark:
· log in to save