GK 2018 Q13
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
Grey Kangaroo 2018, Question 13. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D 41
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.
Solution © UK Mathematics Trust, from the GK 2018 solutions.