Problem 14 - Elementary and Intermediate Algebra

Question

Label each of the following numbers as prime, composite, or neither. $$37$$

Step-by-Step Solution

Answer

37 is a prime number.

Step 1: Understand Prime Numbers

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

Step 2: Understand Composite Numbers

A composite number is a natural number greater than 1 that has more than two positive divisors.

Step 3: Check Divisibility for 37

Test if 37 can be divided by any number other than 1 and 37 itself without leaving a remainder. Start with prime numbers smaller than 37. Check 2, 3, 5, and so on.

Step 4: Determine Result

Since 37 is not divisible by any number other than 1 and 37, it is a prime number.

Key Concepts

Composite Numbers

In mathematics, numbers can be categorized in several ways. One important category is 'composite numbers'. Composite numbers are natural numbers greater than 1 that have more than two positive divisors. This means these numbers can be divided evenly by numbers other than just 1 and themselves.
For example, the number 4 is a composite number because its divisors are 1, 2, and 4. Similarly, 6 is composite because it can be divided by 1, 2, 3, and 6.
Understanding composite numbers is helpful in many math problems, especially in factoring large numbers or finding least common multiples (LCMs). To quickly identify a composite number, try dividing it by smaller prime numbers like 2, 3, 5, etc.

Divisibility

Divisibility is a key concept in number theory and helps determine if one number can be evenly divided by another. If a number can be divided by another without leaving a remainder, it means it's divisible by that number.
For instance, 10 is divisible by 2 (10 ÷ 2 = 5) and by 5 (10 ÷ 5 = 2). When checking for divisibility, we often start with small prime numbers.
Using divisibility rules makes this process easier. For example:

• A number is divisible by 2 if it ends in an even digit.
• A number is divisible by 3 if the sum of its digits is divisible by 3.
• A number is divisible by 5 if it ends in 0 or 5.

These rules speed up the process of finding factors and identifying whether a number is prime or composite.

Natural Numbers

Natural numbers are a basic building block in mathematics. They include all positive integers, starting from 1 and going on indefinitely: 1, 2, 3, 4, and so on. These are the numbers we often use in everyday counting and ranging into more complex mathematical problems.
Natural numbers have specific properties and are used in various mathematical operations, including addition, subtraction, multiplication, and division. They can be either prime or composite, or even neither (like 1, which is unique as it is neither prime nor composite).
To recap, natural numbers are:

• Used for counting.
• Always positive.
• Do not include fractions or decimals.

This simple yet powerful set of numbers forms the foundation for broader mathematical concepts and provides a base for understanding other types of numbers, such as rational and irrational numbers.