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CEMC Gauss (Grade 8), 2020

Algebra Difficulty 1.4 Junior Find the answer Canada

Vita picks a number from 1 to 10. Balil adds 5 to this number and calls his result $b$ . Cali subtracts 5 from Vita’s number and calls her result $c$ . The value of $b-c$ is

Pick one

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.

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