Chapter 4.4, Problem 30WE

To determine

To find:the two perfect integer less than $30$

Answer to Problem 30WE

  $6$ and $28$ are the two perfect integers less than $30$

Explanation of Solution

Given information:

There are two perfect integer less than $30$

Formula used:

A positive integer n is perfect if it is the sum of all its positive factors except n itself.

Calculation:

As per the problem,

A prime number cannot be perfect as it has only two positive factors, $1$ and itself.

Write down the set of factors of all numbers less than $30$ except the prime numbers.

The factor set of $1$ is $1$

The factor set of $4$ is $\{1,2,4\}$

The factor set of $6$ is $\{1,2,3,6\}$

The factor set of $8$ is $\{1,2,4,8\}$

The factor set of $9$ is $\{1,3,9\}$

The factor set of $10$ is $\{1,2,5,10\}$

The factor set of $12$ is $\{1,2,3,4,6,12\}$

The factor set of $14$ is $\{1,2,7,14\}$

The factor set of $15$ is $\{1,3,5,15\}$

The factor set of $16$ is $\{1,2,4,8,16\}$

The factor set of $18$ is $\{1,2,3,6,9,18\}$

The factor set of $20$ is $\{1,2,4,5,10,20\}$

The factor set of $21$ is $\{1,3,7,21\}$

The factor set of $22$ is $\{1,2,11,22\}$

The factor set of $24$ is $\{1,2,3,4,6,8,12,24\}$

The factor set of $25$ is $\{1,5,25\}$

The factor set of $26$ is $\{1,2,13,26\}$

The factor set of $27$ is $\{1,3,9,27\}$

The factor set of $28$ is $\{1,2,4,7,14,28\}$

Of all the integers listed above, on adding the factors of $6$ (except for $6$ itself) and that of $28$ (except of $28$ itself), the result is the integer itself.

Sum of factors of $6$ (except for $6$ itself) $=1+2+3=6$

Sum of factors of $28$ (except of $28$ itself) $=1+2+4+7+14=28$

Hence, $6$ and $28$ are the perfect integers less than $30$