What is the arithmetic mean of the first ten composite numbers?
11.2
Understanding Composite Numbers
Before finding the arithmetic mean, let's understand what composite numbers are. A composite number is a positive integer greater than 1 that has at least one divisor other than 1 and itself. In other words, a composite number can be formed by multiplying two smaller positive integers.
Numbers that are not composite are either prime numbers (which only have two distinct positive divisors: 1 and the number itself) or the number 1 (which is neither prime nor composite).
Identifying the First Ten Composite Numbers
We need to list the positive integers starting from 2 and identify the first ten composite ones:
- 2: Prime (divisors: 1, 2)
- 3: Prime (divisors: 1, 3)
- 4: Composite (divisors: 1, 2, 4) - This is the 1st
- 5: Prime (divisors: 1, 5)
- 6: Composite (divisors: 1, 2, 3, 6) - This is the 2nd
- 7: Prime (divisors: 1, 7)
- 8: Composite (divisors: 1, 2, 4, 8) - This is the 3rd
- 9: Composite (divisors: 1, 3, 9) - This is the 4th
- 10: Composite (divisors: 1, 2, 5, 10) - This is the 5th
- 11: Prime (divisors: 1, 11)
- 12: Composite (divisors: 1, 2, 3, 4, 6, 12) - This is the 6th
- 13: Prime (divisors: 1, 13)
- 14: Composite (divisors: 1, 2, 7, 14) - This is the 7th
- 15: Composite (divisors: 1, 3, 5, 15) - This is the 8th
- 16: Composite (divisors: 1, 2, 4, 8, 16) - This is the 9th
- 17: Prime (divisors: 1, 17)
- 18: Composite (divisors: 1, 2, 3, 6, 9, 18) - This is the 10th
So, the first ten composite numbers are: 4, 6, 8, 9, 10, 12, 14, 15, 16, and 18.
Calculating the Sum of the First Ten Composite Numbers
To find the arithmetic mean, we first need to sum these ten numbers:
Sum = $4 + 6 + 8 + 9 + 10 + 12 + 14 + 15 + 16 + 18$
Let's add them up:
$4 + 6 = 10$
$10 + 8 = 18$
$18 + 9 = 27$
$27 + 10 = 37$
$37 + 12 = 49$
$49 + 14 = 63$
$63 + 15 = 78$
$78 + 16 = 94$
$94 + 18 = 112$
The sum of the first ten composite numbers is 112.
Finding the Arithmetic Mean
The arithmetic mean is calculated by dividing the sum of the numbers by the count of the numbers.
Arithmetic Mean = $\frac{\text{Sum of numbers}}{\text{Number of numbers}}$
In this case, the sum is 112, and there are 10 numbers.
Arithmetic Mean = $\frac{112}{10}$
Arithmetic Mean = $11.2$
Detailed Solution Summary
Here is a summary of the steps:
- Identify the first ten composite numbers: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18.
- Calculate the sum of these numbers: $4 + 6 + 8 + 9 + 10 + 12 + 14 + 15 + 16 + 18 = 112$.
- Divide the sum by the count (which is 10) to find the arithmetic mean: $\frac{112}{10} = 11.2$.
The arithmetic mean of the first ten composite numbers is 11.2.
| Step | Description | Result |
|---|---|---|
| 1 | Identify the first 10 composite numbers | 4, 6, 8, 9, 10, 12, 14, 15, 16, 18 |
| 2 | Calculate the sum | 112 |
| 3 | Calculate the arithmetic mean (Sum / Count) | $112 / 10 = 11.2$ |
Revision Table: Key Concepts
| Term | Definition | Example |
|---|---|---|
| Composite Number | A positive integer > 1 with more than two divisors (1 and itself). | 4, 6, 8, 9, 10, etc. |
| Prime Number | A positive integer > 1 with exactly two distinct positive divisors (1 and itself). | 2, 3, 5, 7, 11, etc. |
| Arithmetic Mean | The sum of a set of numbers divided by the count of the numbers. Also known as average. | Mean of 2, 3, 4 is $\frac{2+3+4}{3} = \frac{9}{3} = 3$ |
Additional Information: Numbers and Their Properties
Understanding different types of numbers like composite numbers, prime numbers, integers, etc., is fundamental in mathematics. The concept of arithmetic mean (average) is widely used in various fields, from statistics to everyday calculations.
- The number 1 is a unique positive integer that is neither prime nor composite.
- The smallest composite number is 4.
- All even numbers greater than 2 are composite because they are divisible by 2 (besides 1 and themselves).
- Odd numbers can be either prime or composite (e.g., 3 is prime, 9 is composite).
Calculating the arithmetic mean helps find a central value for a set of data. It's a simple yet powerful statistical tool.
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