Question

A practical number is a positive integer n such that all positive integers smaller than n can be expressed as sums of distinct divisors of n. For example, 8 is a practical number because the numbers 1 through 7 can be expressed as sums of 1, 2, and 4 as follows: 1 = 1, 2 = 2, 3 = 2 + 1, 4 = 4, 5 = 4 + 1, 6 = 4 + 2, and 7 = 4 + 2 + 1. Show that 18 is a practical number.

Answer 1

The divisors of 18 are

`\mleft\lbrace1,2,3,6,9,18\mright\rbrace`

so we have

`\begin{gathered} 1=1,2=2,3=3,4=3+1,5=3+2,6=6,7=6+1,8=6+2,9=9 \\ 10=1+3+6,11=2+3+6,12=9+3,13=1+3+9,14=2+3+9 \\ 15=6+9,16=1+6+9,17=2+6+9 \end{gathered}`