PK 2020 Q7
7. The positive integers \(a\), \(b\), \(c\), \(d\) satisfy the equation \(ab = 2cd\).
Which of the following numbers could not be the value of the product \(abcd\)?
- A 50
- B 100
- C 200
- D 450
- E 800
Pink Kangaroo 2020, Question 7. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B 100
Since \(ab = 2cd\), the product \(abcd = 2cd \times cd = 2(cd)^2\), hence it must be twice a perfect square. This is true for all the options, except 100 since \(100 = 2 \times 50\) but 50 is not a perfect square. [\(50 = 2 \times 5^2\); \(200 = 2 \times 10^2\); \(450 = 2 \times 15^2\); and \(800 = 2 \times 20^2\).]
Solution © UK Mathematics Trust, from the PK 2020 solutions.