PK 2017 Q17
17. Niall's four children have different integer ages under 18. The product of their ages is 882.
What is the sum of their ages?
- A 23
- B 25
- C 27
- D 31
- E 33
Pink Kangaroo 2017, Question 17. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D 31
The prime factor decomposition of 882 is \(2 \times 3^2 \times 7^2\). The ages must be under 18, so cannot be \(3 \times 7 = 21\) or \(7 \times 7 = 49\). Hence, the only way to create two different numbers using 7 are: 7 and \(2 \times 7 = 14\). This leaves only \(3^2\) which can create the two ages 1 and 9. The sum of the ages is then 1 + 9 + 7 + 14 = 31.
Solution © UK Mathematics Trust, from the PK 2017 solutions.