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Chapter 3: Problem 3

Determine whether the number is prime, composite, or neither. $$89$$

Short Answer

Expert verified
Explain. Answer: The number 89 is a prime number because it is a whole number greater than 1 and has only two factors: 1 and itself (89). It does not have any additional factors, making it a prime number.

Step by step solution

01

Understand the definitions

Recall the definitions of prime, composite, and neither numbers: - A prime number: a whole number greater than 1 that is divisible only by itself and 1. - A composite number: a whole number greater than 1 that has factors other than itself and 1. - Neither: a number is considered neither if it is not prime or composite.
02

Finding factors

To find the factors of 89, test all whole numbers from 1 to the square root of 89 (approximately 9.43), inclusive, and see if they divide 89 evenly without leaving a remainder: 1. 89 梅 1 = 89 (no remainder) 2. 89 梅 2 = 44 remainder 1 3. 89 梅 3 = 29 remainder 2 4. 89 梅 4 = 22 remainder 1 5. 89 梅 5 = 17 remainder 4 6. 89 梅 6 = 14 remainder 5 7. 89 梅 7 = 12 remainder 5 8. 89 梅 8 = 11 remainder 1 9. 89 梅 9 = 9 remainder 8
03

Classify the number

Based on the factors found, we can determine that 89 has only two factors: 1 and 89. Since it is a whole number greater than 1 and has no factors other than itself and 1, we can classify 89 as a prime number.

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Key Concepts

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

Composite Numbers
Composite numbers are fascinating and play a crucial role in the world of mathematics. Unlike prime numbers, composite numbers have more than just two factors. In simple terms, a composite number is a whole number greater than 1 that can be divided evenly by numbers other than 1 and itself.
  • Examples of composite numbers include 4, 6, 8, 9, and 10, as these can be divided evenly by numbers other than 1 and themselves.
  • For instance, the number 4 can be divided by 1, 2, and 4. Here, 2 is the factor that proves it is a composite number.
  • Representing composite numbers with factors such as 1, 2, and 4 for the number 4 makes the concept more understandable.

Understanding composite numbers is important because it helps in determining the properties of numbers and aids in various mathematical operations like factoring and finding least common multiples.
Factors
Factors are fundamental building blocks in mathematics. Factors of a number are any integers that can be multiplied together to get that number.
Knowing how to find factors is crucial when working with both prime and composite numbers.
  • A factor must divide the number completely with no remainder.
  • For example, factors of 12 include 1, 2, 3, 4, 6, and 12.
  • Understanding factors allows us to categorize numbers and understand their divisibility.

Identifying factors is especially useful when simplifying mathematical expressions and solving equations. Factors help us break down numbers into more manageable parts, making math both easier and more intriguing!
Divisibility
Divisibility refers to the ability of one number to be evenly divided by another, meaning there is no remainder left. Learning divisibility is essential as it forms the basis for understanding concepts like prime numbers and simplifying fractions.
  • A number is divisible by another number when, after division, the result is a whole number with a remainder of zero.
  • For instance, 15 is divisible by 3 because 15 divided by 3 equals 5 with no remainder.
  • Divisibility rules help quickly determine whether smaller numbers evenly divide larger numbers.

Applying divisibility skills helps not only with identifying prime and composite numbers but also simplifies many arithmetic operations. These rules are very handy in problems involving factors and multiples, making them an invaluable tool in a mathematician's toolkit.

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Most popular questions from this chapter

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