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Chapter 2: Problem 424

In the following exercises, identify each number as prime or composite. $$ 219 $$

Short Answer

Expert verified
219 is a composite number.

Step by step solution

01

Understand Prime and Composite Numbers

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.
02

Check Small Prime Divisors

To determine if 219 is prime or composite, check if it has any divisors among the smallest prime numbers: 2, 3, 5, 7, 11, 13, etc.
03

Test Divisibility by 2

219 is not even, so it is not divisible by 2.
04

Test Divisibility by 3

Add the digits of 219: 2 + 1 + 9 = 12. Since 12 is divisible by 3, 219 is also divisible by 3.
05

Find Factors

Perform the division to find the factor: 219 梅 3 = 73. So, 219 can be written as 3 脳 73.
06

Conclusion

Since 219 has divisors other than 1 and itself (specifically 3 and 73), it is a composite number.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Prime Numbers
To understand prime numbers, remember that they are numbers greater than 1 that have only two positive divisors: 1 and the number itself. For example, 2, 3, 5, and 7 are all prime numbers because they cannot be divided evenly by any other numbers besides 1 and themselves. If a number can be divided evenly by other numbers, it's not a prime number. Checking for prime numbers is a fundamental concept in number theory and helps improve your understanding of divisibility.
Composite Numbers
Composite numbers are those that have more than two positive divisors. This means there is at least one additional number other than 1 and the number itself that can divide into it evenly. For instance, 4, 6, 8, and 9 are all composite numbers. This is because 4 can be divided by 2, 6 by 2 and 3, and 8 by 2 and 4. Knowing how to identify composite numbers is essential in breaking down numbers into their prime factors, useful in many areas of mathematics.
Divisibility Rules
Divisibility rules are shortcuts that help you determine if a number can be divided by another number without performing the actual division. For example, a number is divisible by 2 if it ends in an even digit like 0, 2, 4, 6, or 8. Another common rule is for 3: a number is divisible by 3 if the sum of its digits is divisible by 3. In the exercise, we used these rules to determine that 219 is divisible by 3 because the sum of its digits (2 + 1 + 9) equals 12, which is divisible by 3. Understanding and applying these rules can make working with large numbers much simpler and faster.

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