19. A list is made of every digit that is the units digit of at least one prime number. How many of the following numbers appear in the list?
A 1
B 2
C 3
D 4
E 5
Solution: D
The numbers 2,3 and 5 are themselves prime numbers so they occur in this list. Also 1 is the units digit of the prime number 11 (and also 31,41 and many more). However a number with units digit 4 is an even number greater than 2 and so is not prime. Thus, there are 4 numbers in the list, namely $1,2,3$ and 5 which are the units digits of at least one prime number.

The German mathematician Peter Dirichlet (1805-1855) proved a theorem from which it follows that there are infinitely many prime numbers in the sequence $1,11,21,31,41, \ldots$ of numbers which have 1 as their units digit.