Problem 62 - Fundamentals of Mathematics

Question

Determine which of the whole numbers are prime and which are composite. 4,575

Step-by-Step Solution

Answer

4,575 is a composite number.

Step 1: Understanding Whole Numbers

Whole numbers are numbers without any fractions or decimals. The provided number, 4,575, is clearly a whole number.

Step 2: Defining Prime and Composite Numbers

A prime number is a whole number greater than 1 that only has two factors: 1 and itself. A composite number has more than two factors.

Step 3: Testing for Factors

To determine if 4,575 is a prime or composite number, we need to find its factors. We start by checking divisibility by smaller prime numbers.

Step 4: Divisibility by 2

A number is divisible by 2 if it is even. 4,575 ends in 5, which is not even, so it is not divisible by 2.

Step 5: Divisibility by 3

A number is divisible by 3 if the sum of its digits is divisible by 3. The sum of the digits of 4,575 is 4 + 5 + 7 + 5 = 21, which is divisible by 3. Therefore, 4,575 is divisible by 3.

Step 6: Conclusion from Divisibility by 3

Since 4,575 is divisible by 3, it means it has more factors than just 1 and itself. It is therefore a composite number.

Key Concepts

Divisibility Rules

Divisibility rules are shortcuts that help us determine whether one number is a factor of another without performing long division. Knowing these rules makes it easy to find factors of a number.
For example:

• A number is divisible by 2 if it ends in an even digit (0, 2, 4, 6, or 8).
• 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 a 0 or 5.

By using these rules, you can quickly identify certain factors. In our original exercise, we used the rule for 3 to find that 4,575 is divisible by 3 because the sum of its digits (4 + 5 + 7 + 5) equals 21, which is divisible by 3. Understanding these rules helps us determine if numbers are composite by identifying more than two factors.

Whole Numbers

Whole numbers are simple and fundamental in mathematics. They include all non-negative numbers without any fractions or decimals, starting from zero and going upwards (0, 1, 2, 3, ...).

Whole numbers are the building blocks for more complex numbers. They are integral in arithmetic and are often used when counting objects. For example, the number 4,575 is a whole number because it does not have any decimal or fractional part. Understanding whole numbers is crucial when discussing topics like prime and composite numbers, as these classifications are made only among whole numbers greater than 1.

Factors of Numbers

Factors are numbers that divide another number completely without leaving a remainder. Finding factors is essential when identifying whether numbers are prime or composite. Prime numbers have exactly two factors: 1 and themselves, while composite numbers have more than two factors.

To determine the factors of a number, one can test divisibility using various rules. For instance, with the number 4,575, checking its divisibility by smaller prime numbers like 2, 3, and 5 helps establish its factors. Since it was found to be divisible by 3, it confirms that 4,575 has factors other than 1 and itself, making it a composite number.

Factors are crucial in many mathematical contexts, such as simplifying fractions or finding the greatest common divisor (GCD) of numbers.