Problem 41 - Basic College Mathematics

Question

Determine whether each number is prime, composite, or neither. $$ 11 $$

Step-by-Step Solution

Answer

11 is a prime number.

Step 1: Define Prime, Composite, and Neither

A prime number is a number greater than 1 that has no positive divisors other than 1 and itself. A composite number is a number greater than 1 that has more than two positive divisors. A number is neither if it does not fit into these categories.

Step 2: Check Divisibility

To determine if 11 is a prime number, check its divisibility. Begin with numbers other than 1 and 11. Check divisibility by 2, 3, 5, and so on.

Step 3: Check Divisibility by 2

The number 11 is odd, so it is not divisible by 2.

Step 4: Check Divisibility by 3

Sum the digits of 11 (1 + 1 = 2). The sum is not divisible by 3, so 11 is not divisible by 3.

Step 5: Check Divisibility by Greater Numbers

For numbers greater than 3, we calculate the square root of 11, which is approximately 3.31. The next prime to check is 5. Since 5 is greater than the square root of 11, further checks for divisibility aren't necessary.

Step 6: Conclusion

Since 11 is only divisible by 1 and 11, it is a prime number.

Key Concepts

Composite Numbers

Composite numbers are fascinating because they hold more than just the factors of 1 and themselves. In essence, they are the opposite of prime numbers. A composite number is any whole number greater than 1 that has at least one other factor besides 1 and itself.

For instance:

• 4 is composite because it is divisible by 1, 2, and 4.
• 15 is composite because it is divisible by 1, 3, 5, and 15.

Understanding composite numbers is essential as it helps detect patterns in factors and enhances the grasp of number theory. By recognizing composites, you also improve your skills in factoring larger numbers and solving related problems.

Remember: If a number can be factored into two smaller whole numbers, it is composite.

Divisibility Rules

Divisibility rules are shortcuts that help determine if one number divides evenly into another without performing lengthy division. They are particularly useful in identifying prime and composite numbers quickly.

Here are some common divisibility rules:

• Divisibility by 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8).
• Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
• Divisibility by 5: A number is divisible by 5 if its last digit is 0 or 5.
• Divisibility by 7: Double the last digit, subtract it from the rest of the number, and if the result is divisible by 7, then so is the number.

These rules simplify many problems and are essential tools for quickly identifying number properties. For example, while determining if 11 is prime, we used these rules to see if it was divisible by numbers like 2 and 3. Knowing and applying these rules can save significant time in solving similar problems.

Math Definitions

Without solid definitions, discussing mathematical concepts would be nearly impossible. Clear definitions ensure everyone understands the terms being used. Here are some critical definitions relevant to prime and composite numbers:

• Prime Number: A number greater than 1 that has no positive divisors other than 1 and itself. Examples include 2, 3, 5, and 7.
• Composite Number: A number greater than 1 that has more than two positive divisors. Examples include 4, 6, 8, and 9.
• Divisibility: A number A is divisible by another number B if dividing A by B leaves no remainder.
• Factor: A factor of a number is an integer that can be multiplied by another integer to produce the number.

Having these terms at your fingertips makes understanding the nature of numbers easier. They provide a foundation for more advanced concepts in mathematics. So, keep these definitions in mind as you work through exercises!