DEFICIENT NUMBER

Deficient Number

Definition Of Deficient Number

If the sum of all the proper divisors of a number is less than the number itself, then the number is called a Deficient Number.

More About Deficient Number

Deficient number is also called as 'defective number'.
All prime numbers are defective numbers.

Example of Deficient Number

Take an example of 14. The proper divisors of 14 are 1, 2, 7.
Sum of the proper divisors = 1 + 2 + 7 = 10.
10 is smaller than 14.
So, 14 is a defective or deficient number.

Video Examples: Deficient, Abundant and Perfect Numbers

Solved Example on Deficient Number

Ques: Identify the defective number from the following.

Choices:

A. 20
B. 12
C. 6
D. 21
Correct Answer: D

Solution:

Step 1: Sum of the proper divisors of 20 = 1 + 2 + 4 + 5 + 10 = 22
Step 2: Sum of the proper divisors of 12 = 1 + 2 + 3 + 4 + 6 = 16
Step 3: Sum of the proper divisors of 6 = 1 + 2 + 3 = 6
Step 4: Sum of the proper divisors of 21 = 1 + 3 + 7 = 11
Step 5: 20, 12, and 6 are not defective numbers as the sum of all the proper divisors of these numbers are not less than the numbers.
Step 6: As sum of all the proper divisors of 21 is less than the number itself, so 21 is a deficient number.

Quick Summary

Deficient numbers are also called defective numbers.
All prime numbers are deficient numbers because their only proper divisor is 1.
To check if a number is deficient, find the sum of its proper divisors and compare it to the original number.

\[ Let \(\sigma(n)\) be the sum of all proper divisors of \(n\). Then, \(n\) is deficient if \(\sigma(n) < n\). \]

🍎 Teacher Insights

Use visual aids like factor trees to help students identify all the proper divisors of a number. Emphasize the difference between divisors and proper divisors.

🎓 Prerequisites

Divisibility
Factors
Proper Divisors
Summation

Check Your Knowledge

Q1: Which of the following numbers is a deficient number?

A. 12 B. 18 C. 21 D. 28

Q2: What is the sum of the proper divisors of 14?

A. 10 B. 14 C. 24 D. 7

Frequently Asked Questions

Q: Are all prime numbers deficient?
A: Yes, because the only proper divisor of a prime number is 1, and 1 is always less than the prime number itself.

Q: How do I find the proper divisors of a number?
A: List all the numbers that divide the given number evenly, excluding the number itself. These are the proper divisors.