IMC 2019 Q8
8. Three consecutive positive integers less than 20 are, in ascending order, prime, even and triangular.
What is the product of these three integers?
- A 6
- B 60
- C 990
- D 1786
- E 2730
Intermediate Mathematical Challenge 2019, Question 8. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: E 2730
First note that the triangular numbers less than 20 are 1, 3, 6, 10, 15. As we are looking for a sequence of consecutive integers which are respectively prime, even, and triangular, the required triangular number is odd. So it is 3 or 15. It is not 3 as 1 is not prime, but it is 15 as 13 is prime.
So the required product = 13 × 14 × 15 = 2730.
Solution © UK Mathematics Trust, from the IMC 2019 solutions. Longer versions with investigations: Solutions and investigations (PDF).