The number 24 is a composite number that possesses exactly eight factors.
Understanding composite numbers and their factors is a fundamental concept in mathematics. A composite number is a positive integer greater than one that has at least one divisor other than one and itself. In simple terms, they are numbers that are not prime numbers.
Identifying Numbers with a Specific Number of Factors
To determine the number of factors a positive integer has, we can use its prime factorization. If a number N can be expressed as p₁^a₁ p₂^a₂ ... pₖ^aₖ, where p₁, p₂, ..., pₖ are distinct prime numbers and a₁, a₂, ..., aₖ are their respective exponents, then the total number of factors (divisors) of N is given by the product of one more than each exponent: (a₁ + 1)(a₂ + 1)...(aₖ + 1)*.
For a number to have exactly 8 factors, its prime factorization must fit one of the following structures:
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A prime number raised to the power of 7 (p⁷):
- (7 + 1) = 8 factors.
- Example: 2⁷ = 128 (Factors: 1, 2, 4, 8, 16, 32, 64, 128)
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*The product of a prime number raised to the power of 3 and another distinct prime number raised to the power of 1 (p₁³ p₂¹):**
- (3 + 1)(1 + 1) = 4 * 2 = 8 factors.
- Example: 2³ 3¹ = 8 3 = 24 (Factors: 1, 2, 3, 4, 6, 8, 12, 24)
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The product of three distinct prime numbers, each raised to the power of 1 (p₁¹ p₂¹ p₃¹):
- (1 + 1)(1 + 1)(1 + 1) = 2 2 2 = 8 factors.
- Example: 2¹ 3¹ 5¹ = 30 (Factors: 1, 2, 3, 5, 6, 10, 15, 30)
Composite Numbers with 8 Factors: Examples
Based on these structures, here are some of the smallest composite numbers with exactly 8 factors:
- 24 (2³ × 3¹)
- 30 (2¹ × 3¹ × 5¹)
- 40 (2³ × 5¹)
- 42 (2¹ × 3¹ × 7¹)
- 54 (2¹ × 3³)
- 56 (2³ × 7¹)
- 66 (2¹ × 3¹ × 11¹)
- 70 (2¹ × 5¹ × 7¹)
- 78 (2¹ × 3¹ × 13¹)
- 88 (2³ × 11¹)
- 102 (2¹ × 3¹ × 17¹)
- 104 (2³ × 13¹)
- 105 (3¹ × 5¹ × 7¹)
- 128 (2⁷)
Understanding "8 as a Factor" vs. "8 Factors"
It's important to distinguish between a number having "8 factors" (meaning eight divisors in total) and a number having "8 as a factor" (meaning it is divisible by 8). Some composite numbers are notable for being divisible by 8. For instance, numbers such as 16 and 24 are examples of composite numbers that are multiples of 8.
Let's examine these two numbers:
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The number 16:
- It is a composite number (factors: 1, 2, 4, 8, 16).
- It is indeed divisible by 8, meaning 8 is one of its factors.
- However, 16 has only 5 factors in total, not 8.
-
The number 24:
- It is a composite number (factors: 1, 2, 3, 4, 6, 8, 12, 24).
- It is divisible by 8, meaning 8 is one of its factors.
- Crucially, 24 has precisely 8 factors in total.
Therefore, 24 stands out as a composite number that not only includes 8 among its factors but also has exactly 8 factors overall, making it a perfect answer to the question.
Summary of Examples
The following table illustrates various composite numbers and their factor counts:
| Number | Prime Factorization | Factors | Number of Factors | Notes |
|---|---|---|---|---|
| 16 | 2⁴ | 1, 2, 4, 8, 16 | 5 | Composite, has 8 as a factor, but not 8 total factors. |
| 24 | 2³ × 3¹ | 1, 2, 3, 4, 6, 8, 12, 24 | 8 | Composite, has 8 as a factor, and has 8 total factors. |
| 30 | 2¹ × 3¹ × 5¹ | 1, 2, 3, 5, 6, 10, 15, 30 | 8 | Composite, has 8 total factors (but not 8 as a factor). |
| 128 | 2⁷ | 1, 2, 4, 8, 16, 32, 64, 128 | 8 | Composite, has 8 as a factor, and has 8 total factors. |
By understanding the rules of prime factorization and factor counting, we can systematically identify any number with a specified number of factors.
