Problem 25 - Algebra 1

Question

Use an indirect proof to prove that the conclusion is true. If \(p\) is an integer and \(p^{2}\) is divisible by \(2,\) then \(p\) is divisible by \(2 .\) (Hint: An odd number can be written as \(2 n+1,\) where \(n\) is an integer. An even number can be written as \(2 n .\) )

Step-by-Step Solution

Answer

By proving a contradiction, we can conclude that if \(p\) is an integer and \(p^{2}\) is divisible by \(2,\) then \(p\) must be divisible by \(2 .\)

Step 1: Understand properties of even and odd numbers

In mathematics, an even number is an integer of the form \(2n,\) where \(n\) is an integer, and an odd number can be expressed as \(2n+1 .\) If a number \(p\) is divisible by \(2,\) it is an even number. If \(p^{2}\) is divisible by \(2,\) then \(p^{2}\) is also an even number.

Step 2: Assume the opposite

To prove by contradiction, start by assuming the opposite of what we're trying to prove. Here, we assume that \(p\) is not divisible by \(2 ,\) meaning that \(p\) is an odd integer. Express \(p\) as \(p = 2n+1 ,\) where \(n\) is an integer.

Step 3: Show that this leads to a contradiction

Next, calculate \(p^{2}\) under the assumption that \(p\) is an odd integer. So, \(p^{2} = (2n+1)^{2} = 4n^{2} + 4n + 1 .\) As we can see, \(p^{2}\) is an odd number, but we were given that \(p^{2}\) is divisible by \(2,\) meaning it's supposed to be an even number. This is a contradiction.

Key Concepts

Even and Odd Numbers

Even and odd numbers are foundational concepts in number theory. Let's make them clear and simple. Even numbers are integers that can be divided by 2 without leaving a remainder. In formula form, an even number looks like this: \(2n\), where \(n\) is any integer. For example, 4, 6, and 10 are even numbers because they can be exactly divided by 2. Odd numbers, on the other hand, cannot be divided by 2 evenly. They have a remainder of 1 when divided by 2. Odd numbers can be expressed as \(2n + 1\). Examples include 3, 5, and 11. Remembering these expressions for even and odd numbers helps in solving many math problems, including proofs by contradiction.

Proof by Contradiction

This is a clever method used in mathematics called 'proof by contradiction' to demonstrate the truth of a statement. The idea is to assume the opposite of what you're trying to prove is true. If this assumption leads to a logical contradiction, then your original statement must be true. Let's consider our exercise. We want to prove that if \(p^2\) is divisible by 2, then \(p\) must be divisible by 2. You begin by assuming the opposite: that \(p\) is not divisible by 2, meaning \(p\) is odd. By exploring the logical implications of this assumption, if \(p\) were actually odd, its square \(p^2\) would also be odd due to the expression \((2n + 1)^2 = 4n^2 + 4n + 1\). But this would contradict the given information that \(p^2\) is divisible by 2. Thus, the opposite assumption fails, proving the original statement.

Divisibility

Divisibility is the concept of one number being evenly divisible by another without any remainder. For example, a number is divisible by 2 if you can split it exactly into two equal parts. Recognizing divisibility is critical in number theory, as it helps determine certain properties of numbers and their relationships. In this exercise, we are concerned with the divisibility of \(p^2\) by 2. If it's indeed divisible by 2, then \(p^2\) is even, implying \(p\) itself must also be even. This is because multiplying an odd number (\(2n + 1\)) results in an odd result, as demonstrated in our proof. Hence, knowing whether a number is divisible can help unravel many questions about integers.

Integer Properties

Integers are numbers without fractional or decimal parts. They can be positive, negative, or zero. Fundamental properties of integers include being whole numbers, meaning they don't possess any fractions or decimals. This makes them particularly useful in discrete mathematics where only whole values make sense. In the problem, the integer \(p\) represents a whole number. We use its characteristics, such as its even or odd properties, when conducting proofs like those involving divisibility or mathematical contradictions. Understanding these integer properties allows us to apply logical reasoning methods like proof by contradiction more effectively, providing clarity in mathematical solutions.