IMC 2019 Q7
7. How many positive cubes less than 5000 end in the digit 5?
- A 1
- B 2
- C 3
- D 4
- E 5
Intermediate Mathematical Challenge 2019, Question 7. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B 2
For the cube of a positive integer to end in a 5, the integer itself must end in a 5. We note that \(5^3 = 125\) and that \(15^3 = 3375\). However \(25^3 > 20^3 = 8000\) and so \(25^3 > 5000\).
Therefore there are only two positive cubes less than 5000 which end in the digit 5, namely 125 and 3375.
Solution © UK Mathematics Trust, from the IMC 2019 solutions. Longer versions with investigations: Solutions and investigations (PDF).