Chapter 5 - Number Theory and the Real Number System - 5.1 Number Theory: Prime and Composite Numbers - Exercise Set 5.1 - Page 258: 125

Deficient numbers are defined as the sum of the divisors except number is less than the number. All the prime numbers are called deficient number and the odd numbers that have one or two distinct factors are prime numbers. Abundant numbers are defined as the sum of divisors except number is greater than that number. And, the abundance can be determined by using the formula as shown below: \[\text{abundance}=\left( \text{sum of all divisors with number}-2\times \text{number} \right)\]

Deficient number: Deficient numbers are defined as the sum of the divisors except number is less than the number. All the prime numbers are called deficient number and the odd numbers that have one or two distinct factors are prime numbers. The deficiency of the deficient numbers can be determined as: \[\text{deficiency}=\left( 2\times \text{number}-\text{sum of all divisors with number} \right)\] For examples, consider the numbers as \[1,2,3,4,5,7,8,11,13,14,15......\] Now, find the divisors of the number\[4\]. Then, \[4=1,2,4\] So, the sum of divisors except number is, \[\begin{align} & S=1+2 \\ & =3 \\ & S<4 \end{align}\] Then, the deficiency of the number\[4\]. \[\begin{align} & \text{deficiency}=\left( 2\times 4-7 \right) \\ & =8-7 \\ & =1 \end{align}\] Find the divisors of the number\[15\]. Then, \[15=1,3,5,15\] So, the sum of divisors except number is, \[\begin{align} & S=1+3+5 \\ & =9 \\ & S<15 \end{align}\] Then, the deficiency of the number \[15\] is, \[\begin{align} & \text{deficiency}=\left( 2\times 15-24 \right) \\ & =30-24 \\ & =6 \end{align}\] Abundant number: Abundant numbers are defined as the sum of divisors except number is greater than that number. And, the abundance can be determined by using the formula as shown below: \[\text{abundance}=\left( \text{sum of all divisors with number}-2\times \text{number} \right)\] The number that has abundance \[1\]is called a quasi perfect number and if the number is not semiperfect then the number is called weird number. The numbers multiple of \[6\]and \[20\]are called abundant numbers. For examples, consider abundant numbers as \[12,18,20,24.....\] Now, find the divisors of the number\[12\]. Then, \[12=1,2,3,4,6,12\] So, the sum of divisors except number is, \[\begin{align} & S=1+2+3+4+6 \\ & =16 \\ & S>12 \end{align}\] Then, the deficiency of the number\[4\]. \[\begin{align} & \text{abundance}=\left( 28-2\times 12 \right) \\ & =28-24 \\ & =4 \end{align}\] Find the divisors of the number\[18\]. Then, \[18=1,2,3,6,9,18\] So, the sum of divisors except number is, \[\begin{align} & S=1+2+3+6+9 \\ & =21 \\ & S>21 \end{align}\] Then, the deficiency of the number\[4\]. \[\begin{align} & \text{abundance}=\left( 40-2\times 19 \right) \\ & =40-38 \\ & =2 \end{align}\]