Problem 23 - Contemporary Precalculus

Question

If the given statement is true, prove it. If it is false, give a counterexample. Every odd positive integer is prime.

Step-by-Step Solution

Answer

Provide a reason for your answer. The statement "Every odd positive integer is prime" is false. This is because we found a counterexample, which is the number 9. Although 9 is an odd positive integer, it is not prime because it has three divisors (1, 3, and 9) and is therefore a composite number.

Step 1: Understand the definition of odd numbers and prime numbers

An odd number is any integer that cannot be exactly divided by 2, meaning it has a remainder of 1 when divided by 2. A prime number is any number greater than 1 that has exactly two distinct positive divisors: 1 and itself.

Step 2: Find a counterexample

To disprove the statement "Every odd positive integer is prime," we need to find at least one example of an odd positive integer that is not prime. Let's check a few odd positive integers and see if they are prime: 3 is odd and prime (divisors: 1, 3) 5 is odd and prime (divisors: 1, 5) 7 is odd and prime (divisors: 1, 7) However, 9 is odd, but it is not prime because it has divisors 1, 3, and 9. So, it is a composite number. Thereby, we have found a counterexample.

Step 3: Conclude

Since we found a counterexample (9), we can conclude that the statement "Every odd positive integer is prime" is false.

Key Concepts

Odd Numbers

Odd numbers are integers that cannot be divided evenly by 2. This means when you divide an odd number by 2, you will always have a remainder of 1. For example, 1, 3, 5, and 7 are all odd numbers.
Odd numbers can be expressed in the form of \(2n + 1\), where \(n\) is an integer. This tells us that if you start with any integer, multiply it by 2 and add 1, you will have an odd number.
Understanding odd numbers is simple, since it's all about the divisibility by 2. For example:

• The number 13 is odd because \(13 \div 2 = 6.5\) with a remainder of 1.
• The number 27 is odd for the same reason, as \(27 \div 2 = 13.5\), again leaving a remainder of 1.

Being familiar with which numbers are considered odd helps in distinguishing when examining other mathematical properties, such as identifying prime or composite numbers.

Prime Numbers

Prime numbers are special because they have exactly two positive divisors: 1 and themselves. Unlike other numbers, prime numbers cannot be formed by multiplying two smaller natural numbers together. This makes them fundamental building blocks in mathematics.
For a number to be prime, it must be greater than 1. Very simply put, the smallest prime number is 2, and it's also the only even prime number.
Some examples of prime numbers include:

• 2 (divisors: 1, 2)
• 3 (divisors: 1, 3)
• 5 (divisors: 1, 5)
• 7 (divisors: 1, 7)

It is important to note that as numbers get larger, determining whether they are prime can become more complex and often requires more advanced techniques than simple divisibility rules.

Composite Numbers

A composite number is an integer that has more than two positive divisors. In other words, it can be divided evenly by numbers other than 1 and itself, which means it can be factored into smaller natural numbers. Composite numbers are the opposite of prime numbers.
For instance, take the number 9: it is a composite number because it can be divided evenly by 1, 3, and 9.

• This is different from a prime number, which only has divisors of 1 and itself.
• Other examples of composite numbers include 4 (divisors: 1, 2, 4), 6 (divisors: 1, 2, 3, 6), and 12 (divisors: 1, 2, 3, 4, 6, 12).

In mathematics, identifying composite numbers is important, as it can help us understand the structure and properties of numbers. Finding a counterexample, such as 9 being an odd number but not a prime number, reveals why not all odd numbers are necessarily prime.