## Practical Numbers

\[n\]

is one for which every positive integer \[1 \le r \le n\]

can be written as a sum of positive divisors of \[n\]

. Example: The divisors of 6 ate 1, 2, 3, and 6.1=1

2=2

3=3

4=1+3

5=2+3

6=1+2+3=6

Hence 6 is a practical number.

Example: The divisors of 5 are 1, and 5.

1=1

2=2 but two is not a divisor of 5.

Example: The divisors of

\[2^n\]

are \[1, \; 2, \; 4,..., \; 2^n\]

.Any number

\[a_1\]

less or equal to than \[2^n\]

can be treated in the following way. Divide by the largest \[2^{j_1}\]

less than \[a_1\]

to get a remainder \[a_2\]

and divide the reminder by the largest power of \[2^{j_2}\]

less than \[a_2\]

. Continuing in this way, we eventually have a remainder of 0 or 1, and since 1 is a divisor of \[2^n\]

we can write \[2^n\]

as a sum of all those powers of 2 we divided by, plus either 0 or 1.Hence

\[2^n\]

is a practical number.