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Problem 448

Number theory Difficulty 2.4 Multiple choice CEMC Gauss (Grade 7) · Canada · 2020

The positive divisors of 12 (other than itself) are $1, 2, 3, 4$ , and 6. Their sum, $1+2+3+4+6$ , is greater than 12. An abundant number is a number for which the sum of its positive divisors (other than itself) is greater than the number itself. This means that 12 is an abundant number. Which of the following is also an abundant number?

Pick one

Official solution

In the table below, we list the divisors of each of the given answers (other than the number itself) and determine their sum.

Given Answers	Divisors	Sum of the Divisors
8	$1,2,4$	$1+2+4=7$
10	$1,2,5$	$1+2+5=8$
14	$1,2,7$	$1+2+7=10$
18	$1,2,3,6,9$	$1+2+3+6+9=21$
22	$1,2,11$	$1+2+11=14$

The sum of the divisors of each of $8,10,14$ , and 22 is less than the number itself.

Each of these four answers is not an abundant number. (These numbers are called “deficient”.)

The sum of the divisors of 18 is 21, which is greater than 18.
Thus 18 is an abundant number.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.