Perfect Numbers
A perfect number is a special type of number whose proper factors (all factors except the number itself) add up to the number. Perfect numbers have fascinated mathematicians for thousands of years.
In Class 5, students learn to identify perfect numbers by finding all factors, excluding the number itself, and checking if their sum equals the original number. The two perfect numbers relevant at this level are 6 and 28.
What is Perfect Numbers - Class 5 Maths (Factors and Multiples)?
A perfect number is a positive whole number that equals the sum of all its proper factors (factors excluding the number itself).
Example:
- Factors of 6: 1, 2, 3, 6
- Proper factors (exclude 6): 1, 2, 3
- Sum of proper factors: 1 + 2 + 3 = 6
- Since the sum equals the number, 6 is a perfect number.
Related terms:
- If the sum of proper factors is less than the number → Deficient number (e.g., 8: 1+2+4 = 7 < 8)
- If the sum of proper factors is more than the number → Abundant number (e.g., 12: 1+2+3+4+6 = 16 > 12)
Perfect Numbers Formula
A number N is perfect if: Sum of all proper factors of N = N
Solved Examples
Example 1: Example 1: Verify that 6 is a perfect number
Problem: Show that 6 is a perfect number.
Solution:
Step 1: Find all factors of 6: 1, 2, 3, 6
Step 2: Proper factors (exclude 6): 1, 2, 3
Step 3: Sum = 1 + 2 + 3 = 6
Step 4: Sum = Number ✓
Answer: 6 is a perfect number.
Example 2: Example 2: Verify that 28 is a perfect number
Problem: Show that 28 is a perfect number.
Solution:
Step 1: Find all factors of 28: 1, 2, 4, 7, 14, 28
Step 2: Proper factors: 1, 2, 4, 7, 14
Step 3: Sum = 1 + 2 + 4 + 7 + 14 = 28
Step 4: Sum = Number ✓
Answer: 28 is a perfect number.
Example 3: Example 3: Check if 10 is a perfect number
Problem: Is 10 a perfect number?
Solution:
Step 1: Factors of 10: 1, 2, 5, 10
Step 2: Proper factors: 1, 2, 5
Step 3: Sum = 1 + 2 + 5 = 8
Step 4: 8 ≠ 10. Sum < Number.
Answer: 10 is NOT a perfect number. It is a deficient number.
Example 4: Example 4: Check if 12 is a perfect number
Problem: Is 12 a perfect number?
Solution:
Step 1: Factors of 12: 1, 2, 3, 4, 6, 12
Step 2: Proper factors: 1, 2, 3, 4, 6
Step 3: Sum = 1 + 2 + 3 + 4 + 6 = 16
Step 4: 16 ≠ 12. Sum > Number.
Answer: 12 is NOT a perfect number. It is an abundant number.
Example 5: Example 5: Check if 15 is a perfect number
Problem: Is 15 a perfect number?
Solution:
Step 1: Factors of 15: 1, 3, 5, 15
Step 2: Proper factors: 1, 3, 5
Step 3: Sum = 1 + 3 + 5 = 9
Step 4: 9 ≠ 15.
Answer: 15 is NOT a perfect number. It is deficient.
Example 6: Example 6: Check if 496 is a perfect number
Problem: Verify that 496 is a perfect number.
Solution:
Step 1: Factors of 496: 1, 2, 4, 8, 16, 31, 62, 124, 248, 496
Step 2: Proper factors: 1, 2, 4, 8, 16, 31, 62, 124, 248
Step 3: Sum = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496
Answer: 496 is a perfect number. It is the third perfect number after 6 and 28.
Example 7: Example 7: Classifying numbers
Problem: Classify each number as perfect, deficient, or abundant: 8, 6, 18.
Solution:
- 8: Proper factors: 1, 2, 4. Sum = 7. Since 7 < 8 → Deficient
- 6: Proper factors: 1, 2, 3. Sum = 6. Since 6 = 6 → Perfect
- 18: Proper factors: 1, 2, 3, 6, 9. Sum = 21. Since 21 > 18 → Abundant
Example 8: Example 8: Finding factors systematically
Problem: Find all proper factors of 20 and check if it is perfect.
Solution:
Step 1: Check divisibility from 1 to √20 ≈ 4.5:
- 20 ÷ 1 = 20 → factors: 1, 20
- 20 ÷ 2 = 10 → factors: 2, 10
- 20 ÷ 4 = 5 → factors: 4, 5
Step 2: All factors: 1, 2, 4, 5, 10, 20
Step 3: Proper factors: 1, 2, 4, 5, 10. Sum = 22.
Step 4: 22 > 20 → Abundant, not perfect.
Example 9: Example 9: Word problem
Problem: Arjun has 28 marbles. He wants to divide them into groups where each group size is a factor of 28 (excluding 28 itself). Can the sizes of all such groups add up to exactly 28?
Solution:
Proper factors of 28: 1, 2, 4, 7, 14
Sum = 1 + 2 + 4 + 7 + 14 = 28
Answer: Yes! Since 28 is a perfect number, the sum of all proper group sizes equals 28.
Key Points to Remember
- A perfect number equals the sum of its proper factors (all factors except itself).
- The first four perfect numbers are: 6, 28, 496, 8128.
- A deficient number has proper factor sum less than itself (e.g., 8, 10, 14).
- An abundant number has proper factor sum greater than itself (e.g., 12, 18, 20).
- All even perfect numbers discovered so far follow a pattern involving prime numbers.
- Perfect numbers are rare — there are only 4 below 10,000.
- To check if a number is perfect: find all factors, exclude the number, add the rest, and compare.
Practice Problems
- Is 24 a perfect number? Find the sum of its proper factors.
- Is 8128 a perfect number? (Hint: its proper factors are 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064)
- Classify the following as perfect, deficient, or abundant: 9, 6, 18, 28, 14.
- Find all proper factors of 36 and check if it is perfect.
- The sum of proper factors of a number is 1. What type of number is it?
- List all factors of 496 and verify it is a perfect number.
- Is any prime number a perfect number? Explain why or why not.
Frequently Asked Questions
Q1. What is a perfect number?
A perfect number is a positive whole number that equals the sum of all its proper factors (all factors except the number itself). The smallest perfect number is 6, because 1 + 2 + 3 = 6.
Q2. What are the first few perfect numbers?
The first four perfect numbers are 6, 28, 496, and 8,128. Perfect numbers are very rare — the fifth one is 33,550,336.
Q3. Can a prime number be a perfect number?
No. A prime number has only two factors: 1 and itself. The sum of proper factors is just 1, which is always less than the prime number. So every prime number is deficient.
Q4. What is a deficient number?
A deficient number is one where the sum of its proper factors is less than the number itself. For example, 14 has proper factors 1, 2, 7, and their sum is 10, which is less than 14.
Q5. What is an abundant number?
An abundant number is one where the sum of its proper factors is more than the number itself. For example, 12 has proper factors 1, 2, 3, 4, 6, and their sum is 16, which is greater than 12.
Q6. How do I find proper factors of a number?
Find all factors by checking divisibility from 1 up to the square root of the number. Each divisor gives a pair of factors. Then remove the number itself from the list — the remaining factors are the proper factors.
Q7. Are all even numbers perfect numbers?
No. Most even numbers are not perfect. Among even numbers up to 100, only 6 and 28 are perfect. Many even numbers are deficient (like 2, 4, 8) or abundant (like 12, 18, 20).
Q8. Why are perfect numbers important?
Perfect numbers are studied in number theory and have applications in coding theory and cryptography. For Class 5, they help students practice finding factors and understanding divisibility, and they show that numbers have interesting hidden patterns.
