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Chapter 1: Problem 10

Label each of the following numbers as prime, composite, or neither. $$15$$

Short Answer

Expert verified
15 is a composite number.

Step by step solution

01

- Understand Definitions

Understand the definitions of prime, composite, and neither: A prime number has exactly two distinct positive divisors, 1 and itself. A composite number has more than two distinct positive divisors. 'Neither' applies to numbers like 1, which do not meet the criteria for prime or composite.
02

- Analyze the Number 15

Identify the factors of the number 15. To do this, determine which numbers divide 15 without leaving a remainder.
03

- Find Divisors

List all the divisors: 1, 3, 5, and 15.
04

- Classify the Number

Since the number 15 has more than two distinct positive divisors (1, 3, 5, and 15), it qualifies as 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
Prime numbers are fascinating elements in the realm of mathematics. A number is considered prime if it has exactly two distinct positive divisors: 1 and itself. This means that no other numbers can divide a prime number without leaving a remainder. For example, the number 7 is prime because it can only be divided by 1 and 7. Let's review more examples to solidify our understanding:
  • 3 (divisors: 1 and 3)
  • 11 (divisors: 1 and 11)
  • 23 (divisors: 1 and 23)
Learning to identify prime numbers is a fundamental skill that will help you in many areas of mathematics.
Composite Numbers
Understanding composite numbers is equally important. A composite number has more than two distinct positive divisors. This means that, apart from 1 and itself, other numbers can also divide it without leaving a remainder. To identify a composite number, check if it can be divided evenly by any number other than 1 and itself. For instance, the number 15 is composite because it has four divisors: 1, 3, 5, and 15. Other examples include:
  • 4 (divisors: 1, 2, and 4)
  • 12 (divisors: 1, 2, 3, 4, 6, and 12)
  • 20 (divisors: 1, 2, 4, 5, 10, and 20)
Knowing how to identify composite numbers will aid you in various mathematical tasks, such as simplifying fractions or finding greatest common divisors.
Number Divisors
Number divisors are at the heart of understanding both prime and composite numbers. A divisor, or factor, of a number divides that number without leaving a remainder. For example, the divisors of 15 are 1, 3, 5, and 15. This concept is crucial in many areas of mathematics, including simplifying expressions and solving equations. Here's a step-by-step approach to finding divisors:
  • Start with 1 and the number itself (these are always divisors)
  • Check smaller numbers to see if they divide evenly
  • Continue this process up to the square root of the number
Recognizing all divisors of a number helps in determining whether it is prime or composite.
Factorization
Factorization is the process of breaking down a number into its prime factors. This is an essential skill in mathematics, as it simplifies many complex problems. For instance, factorizing 15 results in 3 and 5 because 15 = 3 脳 5. To factorize a number, follow these steps:
  • Start by dividing the number by the smallest prime (2, 3, 5, etc.)
  • Continue dividing the quotient by prime numbers until you reach 1
  • List all the prime numbers used as factors
Example: Factorize 60.
60 梅 2 = 30
30 梅 2 = 15
15 梅 3 = 5
5 梅 5 = 1.
Thus, the prime factors of 60 are 2, 2, 3, and 5. Mastering factorization will boost your problem-solving skills in algebra, number theory, and beyond.

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