IMC 2013 Q11
11. For which of the following numbers is the sum of all its factors not equal to a square number?
- A 3
- B 22
- C 40
- D 66
- E 70
Intermediate Mathematical Challenge 2013, Question 11. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C 40
We first look at 66 = 2 × 3 × 11. Its factors involve none, one, two or all three of these primes. So the factors are 1, 2, 3, 11, 6, 22, 33, 66; and their sum is 144 = 122. Similarly, we can check that the sum of the factors of 3, 22, 40 and 70 is, respectively, 4 = 22, 36 = 62, 90 and 144 = 122. So 40 is the only alternative for which the sum of the factors is not a square number.
Solution © UK Mathematics Trust, from the IMC 2013 solutions. Longer versions with investigations: Solutions and investigations (PDF).