Problem 11 - Elementary and Intermediate Algebra

Question

Label each of the following numbers as prime, composite, or neither. $$ 31 $$

Step-by-Step Solution

Answer

31 is a prime number.

Step 1: Understand Definitions

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A composite number is a natural number greater than 1 that has more than two positive divisors. A number that is not greater than 1 is classified as neither.

Step 2: Check Divisibility

For the number 31, check if it can be divided evenly by any number other than 1 and 31 itself. This means you need to test the divisibility by numbers such as 2, 3, 5, etc. up to the square root of 31.

Step 3: Test Divisibility by 2

31 is an odd number, so it is not divisible by 2.

Step 4: Test Divisibility by 3

Sum the digits of 31: 3 + 1 = 4. Since 4 is not divisible by 3, 31 is not divisible by 3.

Step 5: Test Divisibility by 5

31 does not end in 0 or 5, so it is not divisible by 5.

Step 6: Confirm Prime Status

Since 31 is not divisible by any number other than 1 and 31, it is a prime number.

Key Concepts

Prime and Composite Numbers

A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. In simpler terms, a prime number has no divisors other than 1 and itself. For example, numbers like 2, 3, 5, 7, and 31 are prime numbers. These numbers can only be divided evenly (leaving no remainder) by 1 and by the number itself.
On the other hand, composite numbers are natural numbers greater than 1 that have more than just 1 and itself as divisors. This means you can find at least one other number that divides the composite number evenly. Examples include 4, 6, 9, 12, and 15. For instance, the number 9 can be divided evenly by 1, 3, and 9, making it a composite number. Understanding these definitions helps us classify numbers correctly and solve related problems.

Divisibility Rules

Divisibility rules are handy shortcuts that help us quickly determine whether a number can be divided evenly by another number without doing complex calculations. Here are some basic rules:

• Divisibility by 2: A number is divisible by 2 if it is even, meaning it ends with 0, 2, 4, 6, or 8.
• Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3. For example, 123 is divisible by 3 because 1 + 2 + 3 = 6, and 6 is divisible by 3.
• Divisibility by 5: A number is divisible by 5 if it ends in 0 or 5.
• Divisibility by 7: A more complex one, where you double the last digit, subtract it from the rest of the number, and if the result is divisible by 7, then so is the original number.

Knowing these rules helps quickly filter out non-prime numbers when checking for prime status. For example, to check if 31 is prime, you test it against these rules and see that it isn’t divisible by 2, 3, or 5. It confirms that there are no smaller natural numbers it divides evenly with, making it a prime number.

Classification of Numbers

Classification of numbers helps in understanding the different categories numbers fall into. Here are some basic classifications:

• Natural Numbers: These are positive integers starting from 1, 2, 3, and so on.
• Whole Numbers: These are all natural numbers including 0. So, 0, 1, 2, 3, etc.
• Integers: These include all whole numbers and their negative counterparts. So, -3, -2, -1, 0, 1, 2, 3, etc.
• Rational Numbers: Numbers that can be expressed as a fraction \(\frac{a}{b}\) where a and b are integers and b is not zero. For example, 1/2, 5/3, and -2/4.
• Real Numbers: Include all rational and irrational numbers (numbers that cannot be expressed as a simple fraction).

Understanding these classifications is crucial in solving math problems and for a deeper understanding of number properties. For instance, knowing 31 is a natural number, an integer, and a prime number helps in various math contexts.