PK 2015 Q14
14. The two roots of the quadratic equation
\(x^2 - 85x + c = 0\)
are both prime numbers. What is the sum of the digits of \(c\)?
- A 12
- B 13
- C 14
- D 15
- E 21
Pink Kangaroo 2015, Question 14. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B 13
Suppose the quadratic has roots \(p\) and \(q\). So it factorises as \((x - p)(x - q)\). This expands to give \(x^2 - (p + q)x + pq\). Comparing this with \(x^2 - 85x + c\) shows that \(p + q = 85\) and \(pq = c\). It follows that one of \(p\), \(q\) is even and one is odd. The only even prime is 2 so the primes \(p\), \(q\) are 2 and 83. Therefore \(c = 2 \times 83 = 166\), and therefore has digit sum 13.
Solution © UK Mathematics Trust, from the PK 2015 solutions.