No, 42 is not a perfect number.
A perfect number is a positive integer that is equal to the sum of its proper positive divisors (divisors excluding the number itself). To determine if 42 fits this definition, we need to find its proper divisors and sum them up.
What Defines a Perfect Number?
A number is considered "perfect" in number theory if the sum of its positive divisors, excluding the number itself, equals the number itself. For instance, the number 6 is a perfect number because its proper divisors are 1, 2, and 3, and their sum (1 + 2 + 3) is 6.
- Key Characteristic: The sum of proper divisors equals the number.
- First Few Perfect Numbers: The smallest perfect numbers are 6, 28, 496, and 8128.
For more in-depth information, you can explore the concept of Perfect Numbers on Wikipedia.
Analyzing 42: A Step-by-Step Calculation
To verify whether 42 is a perfect number, let's list its divisors and calculate their sum.
Step 1: Identify the Divisors of 42
The divisors of 42 are the numbers that divide 42 evenly.
- 1, 2, 3, 6, 7, 14, 21, 42
Step 2: List the Proper Divisors
Proper divisors are all positive divisors of a number, excluding the number itself.
- The proper divisors of 42 are: 1, 2, 3, 6, 7, 14, 21.
Step 3: Sum the Proper Divisors
Now, let's add these proper divisors together:
- 1 + 2 + 3 + 6 + 7 + 14 + 21 = 54
Step 4: Compare the Sum to the Original Number
- The sum of the proper divisors of 42 is 54.
- The number itself is 42.
Since 54 ≠ 42, 42 is not a perfect number.
Classification of 42
Based on the sum of its proper divisors, numbers can be classified into three categories:
| Category | Definition | Example |
|---|---|---|
| Perfect | The sum of its proper divisors is equal to the number. | 6 (1+2+3=6) |
| Abundant | The sum of its proper divisors is greater than the number. | 12 (1+2+3+4+6=16 > 12) |
| Deficient | The sum of its proper divisors is less than the number. | 10 (1+2+5=8 < 10) |
Since the sum of the proper divisors of 42 (which is 54) is greater than 42, 42 is classified as an abundant number.
