Smallest Abundant Number
Weekly Problem 34 - 2017
An abundant number is a positive integer N such that the sum of the factors of N is larger than 2N. What is the smallest abundant number?
An abundant number is a positive integer N such that the sum of the factors of N is larger than 2N. What is the smallest abundant number?
Age
11 to 14
Challenge level
Problem
An abundant number is a positive integer $N$ such that the sum of the factors of $N$ (not including $N$ itself) is greater than $N$.
What is the smallest abundant number?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
Answer: 12
Primes - only factor is 1 (and self) so not abundant
| number | factors | sum | abundant? |
|---|---|---|---|
| 4 | 1,2 | 3 | No |
| 6 | 1,2,3 | 6 | No |
| 8 | 1,2,4 | 7 | No |
| 9 | 1,3 | 4 | No |
| 10 | 1,2,5 | 7 | No |
| 12 | 1,2,3,4,6 | 16 | Yes |
This shows that the smallest abundant number is $12$.
$6$ is called a perfect number, as the sum of its factors is equal to the number.