IMC 2012 Q10
10. An ‘abundant’ number is a positive integer \(N\), such that the sum of the factors of \(N\) (excluding \(N\) itself) is greater than \(N\). What is the smallest abundant number?
- A 5
- B 6
- C 10
- D 12
- E 15
Intermediate Mathematical Challenge 2012, Question 10. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D 12
The table shows the first 12 positive integers, \(N\), and the sum, \(S\), of the factors of \(N\) excluding \(N\) itself. As can be seen, 12 is the first value of \(N\) for which this sum exceeds \(N\), so 12 is the smallest abundant number.
(Note that for \(N = 6\) the sum, \(S\), also equals 6. For this reason, 6 is known as a ‘perfect number’. After 6, the next two perfect numbers are 28 and 496.)
Solution © UK Mathematics Trust, from the IMC 2012 solutions. Longer versions with investigations: Solutions and investigations (PDF).