Hamilton 2016 Q1
1. No digit of the positive integer \(N\) is prime. However, all the single-digit primes divide \(N\) exactly.
What is the smallest such integer \(N\)?
Hamilton Olympiad 2016, Question 1. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: 840
The single-digit primes are 2, 3, 5 and 7. Each of them divides \(N\), so that \(2 \times 3 \times 5 \times 7\) divides \(N\). Written another way, this means that \(N\) is a multiple of \(2 \times 3 \times 5 \times 7 = 210\).
But one of the digits of 210 is the prime 2, so \(N\) is not 210, and one of the digits of \(2 \times 210 = 420\) is also 2, so \(N\) is not 420 either. Furthermore, one of the digits of \(3 \times 210 = 630\) is the prime 3, so \(N\) is not 630. However, none of the digits of \(4 \times 210 = 840\) is a prime, so \(N\) can be 840.
We have ruled out all the smaller options, therefore the smallest possible integer \(N\) is 840.
Solution © UK Mathematics Trust, from the Hamilton Olympiad 2016 solutions.