IMC 2018 Q10
10. Which of these integers cannot be expressed as the difference of two squares?
- A 5
- B 7
- C 8
- D 9
- E 10
Intermediate Mathematical Challenge 2018, Question 10. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: E 10
The difference of two squares \(a^2 - b^2\) factorises to \((a + b)(a - b)\). Therefore for it to be possible to express an integer as the difference of two squares, it must be possible to factorise the integer in the form \((a + b)(a - b)\), where \(a\) and \(b\) are both integers.
Note that \(5 = (3 + 2)(3 - 2)\); \(7 = (4 + 3)(4 - 3)\); \(8 = (3 + 1)(3 - 1)\); \(9 = (5 + 4)(5 - 4)\).
So 5, 7, 8, 9 can all be written as the difference of two squares.
(It is left to the reader to show that 10 cannot be the difference of two squares as it cannot be written in the form \((a + b)(a - b)\) where a and b are both integers.)
Solution © UK Mathematics Trust, from the IMC 2018 solutions. Longer versions with investigations: Solutions and investigations (PDF).