IMC 2008 Q5
5. Which of the following is not equal to a whole number?
- A \(\dfrac{594}{5 + 9 + 4}\)
- B \(\dfrac{684}{6 + 8 + 4}\)
- C \(\dfrac{756}{7 + 5 + 6}\)
- D \(\dfrac{873}{8 + 7 + 3}\)
- E \(\dfrac{972}{9 + 7 + 2}\)
Intermediate Mathematical Challenge 2008, Question 5. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D \(\dfrac{873}{8 + 7 + 3}\)
In each of the fractions, the denominator is 18 and the sum of the digits of the numerator is also 18. So every numerator is a multiple of 9 and the even numerators are also multiples of 18. However, 873 is not a multiple of 18, so \(\dfrac{873}{8 + 7 + 3}\) is the only expression not equal to a whole number.
Solution © UK Mathematics Trust, from the IMC 2008 solutions.