IMC 2011 Q22
22. You are given that \(5^p = 9\), \(9^q = 12\), \(12^r = 16\), \(16^s = 20\) and \(20^t = 25\). What is the value of \(pqrst\) ?
- A 1
- B 2
- C 3
- D 4
- E 5
Intermediate Mathematical Challenge 2011, Question 22. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B 2
\(5^p = 9\) and \(9^q = 12\). Therefore \((5^p)^q = 12\), that is \(5^{pq} = 12\). Similarly, as \(12^r = 16\) then \(5^{pqr} = 16\), as \(16^s = 20\) then \(5^{pqrs} = 20\) and, finally, as \(20^t = 25\) then \(5^{pqrst} = 25\). Therefore \(pqrst = 2\).
Solution © UK Mathematics Trust, from the IMC 2011 solutions. Longer versions with investigations: Solutions and investigations (PDF).