PK 2020 Q11
11. Masha put a multiplication sign between the second and third digits of the number 2020 and noted that the resulting product 20 × 20 was a square number.
How many integers between 2010 and 2099 (including 2020) have the same property?
- A 1
- B 2
- C 3
- D 4
- E 5
Pink Kangaroo 2020, Question 11. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C 3
Each number begins with 20, and \(20 = 5 \times 2^2\). Hence, to make a square product, the last two digits must make a number which is a product of 5 and a square number. The possibilities between 10 and 99 are \(5 \times 2^2 = 20\), \(5 \times 3^2 = 45\) and \(5 \times 4^2 = 80\). Therefore there are three possible numbers: 2020, 2045, 2080.
Solution © UK Mathematics Trust, from the PK 2020 solutions.