H1. No digit of the positive integer $N$ is prime. However, all the single-digit primes divide $N$ exactly.
What is the smallest such integer $N$ ?
Solution
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 .