Sure, let's explore the properties of different types of numbers to find the answer to this question.
### Step-by-Step Solution:
1. Understanding the Options:
- Perfect Number: A number is called a perfect number if the sum of all its divisors, including the number itself, equals twice the number. For example, the factors of 6 (1, 2, 3, 6) add up to 12, which is twice 6.
- Even Number: A number is even if it is divisible by 2. This does not involve summing its factors in any specific way.
- Odd Number: A number is odd if it is not divisible by 2. Again, this property does not relate to the sum of its factors.
- Prime Number: A number is prime if it has exactly two distinct positive divisors: 1 and the number itself. Like even or odd numbers, this does not describe the sum of its factors.
2. Analyzing the Definition:
- We need a number where the sum of its factors equals twice the number itself.
- Perfect numbers are exactly such numbers. The definition of perfect numbers states that if you add all the factors (including the number itself), the total equals twice the number.
3. Examples to Illustrate:
- Consider the number 6:
- Its factors are 1, 2, 3, and 6.
- The sum is 1 + 2 + 3 + 6 = 12.
- 12 is twice 6.
- Consider the number 28:
- Its factors are 1, 2, 4, 7, 14, and 28.
- The sum is 1 + 2 + 4 + 7 + 14 + 28 = 56.
- 56 is twice 28.
By carefully analyzing the definitions and properties above, we can conclude that the correct answer is:
(a) perfect number
