Problem 51 - Fundamentals of Mathematics

Question

Determine which of the whole numbers are prime and which are composite. 9

Step-by-Step Solution

Answer

9 is a composite number.

Step 1: Understand Prime and Composite Numbers

A prime number is a whole number greater than 1 that has no positive divisors other than 1 and itself. A composite number is a whole number greater than 1 that has more than two positive divisors.

Step 2: List Divisors of the Number 9

To determine if 9 is prime or composite, begin by listing all the divisors of 9. These include the numbers that divide 9 without leaving any remainder. The divisors of 9 are 1, 3, and 9.

Step 3: Determine the Type of Number

Since 9 has more than two divisors (1, 3, and 9), it does not meet the criteria to be a prime number, which should only have divisors 1 and itself. Therefore, 9 is a composite number since it has additional divisors (3).

Key Concepts

Divisors

Divisors are numbers that can divide another number without leaving any remainder. If you take a number, say 9, and divide it by different integers, the numbers that divide it completely are called its divisors. In simple terms, when you divide 9 by its divisors, you get a whole number without any fractions or decimals.
For example:

• 9 ÷ 1 = 9 (1 is a divisor of 9)
• 9 ÷ 3 = 3 (3 is a divisor of 9)
• 9 ÷ 9 = 1 (9 is a divisor of 9)

When determining whether a number is prime or composite, identifying its divisors is crucial. A number with only two divisors (1 and itself) is labeled 'prime', while if it has more than two divisors, it falls into the 'composite' category.

Prime Number Definition

A prime number is a whole number greater than 1 that can only be divided by 1 and itself without any remainder. Prime numbers serve as the building blocks of whole numbers because any whole number can be represented as a product of prime numbers.
To check if a number is a prime, you try dividing it by any whole numbers other than 1 and itself. If no such divisors exist, the number is prime.
For example:

• 2 is a prime number. It can only be divided by 1 and 2.
• 5 is a prime number. It can only be divided by 1 and 5.
• 7 is a prime number. It can only be divided by 1 and 7.

Remember, the numbers 0 and 1 are not considered prime.

Composite Number Characteristics

Composite numbers are whole numbers greater than 1 with more than two divisors. Unlike prime numbers, composites can be factored into smaller numbers other than 1 and the number itself. This means you can break them down into simpler multipliers.
For instance:

• 4 is a composite number because it can be divided by 1, 2, and 4.
• 6 is a composite number because it can be divided by 1, 2, 3, and 6.
• 9 is a composite number because it can be divided by 1, 3, and 9.

Composite numbers are essential in math as they highlight the possibility of breaking down numbers into factors, helping in multiplication and division operations.