PK 2017 Q10
10. The sum of the squares of three consecutive positive integers is 770. What is the largest of these integers?
- A 15
- B 16
- C 17
- D 18
- E 19
Pink Kangaroo 2017, Question 10. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C 17
The three squares will be approximately a third of 770, so roughly 250. Adding up \(15^2 = 225\), \(16^2 = 256\) and \(17^2 = 289\) gives the total 770.
Alternatively, using algebra we could let \(n\) be the middle integer and then add the squares \((n - 1)^2 + n^2 + (n + 1)^2 = n^2 - 2n + 1 + n^2 + n^2 + 2n + 1 = 3n^2 + 2 = 770\).
This gives \(n^2 = 256\) and hence \(n = 16\), so the largest integer is 17.
Solution © UK Mathematics Trust, from the PK 2017 solutions.