13. Alice wants to write down a list of prime numbers less than 100 , using each of the digits 1,2 , 3,4 and 5 once and no other digits. Which prime number must be in her list?
A 2
B 5
C 31
D 41
E 53
13. D First note that none of 2,4 or 5 can occur as the final digit of a two-digit prime. Also 21 and 51 are not prime since they are divisible by 3 . Therefore the primes which might occur in Alice's list are 2,3,5,13,23,31, 41, 43 and 53. Since 1 and 4 are not prime, they must occur in Alice's list as digits in some two-digit prime. If 41 is not used, the only two-digit primes which could use the 4 and the 1 are 43 and one of 31 or 13 . However, this would repeat the digit 3. Hence 41 must be in Alice's list together with one of the groups $2,3,5$ or 2,53 or 5,23 .