IMC 2016 Q16
16. How many of the following positive integers are divisible by 24?
\(2^2 \times 3^2 \times 5^2 \times 7^3\) \(2^2 \times 3^2 \times 5^3 \times 7^2\) \(2^2 \times 3^3 \times 5^2 \times 7^2\) \(2^3 \times 3^2 \times 5^2 \times 7^2\)
- A 0
- B 1
- C 2
- D 3
- E 4
Intermediate Mathematical Challenge 2016, Question 16. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B 1
The prime factorisation of \(24 = 2^3 \times 3\). Therefore all multiples of 24 must include both \(2^3\) and 3 in their prime factorisation. Of the options given, only the last includes \(2^3\). As it is also a multiple of 3, it is a multiple of 24.
Solution © UK Mathematics Trust, from the IMC 2016 solutions. Longer versions with investigations: Solutions and investigations (PDF).