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Magazine Chapter 1: Problem 18
Prove that if \(m\) and \(n\) are integers and \(m n\) is even, then \(m\) is even or \(n\) is even.
Short Answer
Expert verified
If m n is even, either m or n must be even.
Step by step solution
01
Understand the problem
You need to prove that if the product of two integers, say m and n, is even, then at least one of the integers is even.
02
Define even and odd integers
Recall that an integer is even if it is divisible by 2, i.e., it can be written as 2k, where k is an integer. An integer is odd if it can be written as 2k + 1.
03
Represent product as even
Given: m n is even. The definition of even implies that m n = 2k for some integer k.
04
Assumption for proving by contradiction
Assume that both m and n are odd. Express m and n as 2a + 1 and 2b + 1 respectively, where a and b are integers.
05
Calculate product of odd integers
If m = 2a + 1 and n = 2b + 1, then m n = (2a + 1)(2b + 1). Expanding this product gives: m n = 4ab + 2a + 2b + 1 = 2(2ab + a + b) + 1.
06
Analyze the product
Notice that m n = 2(2ab + a + b) + 1 is odd since it is in the form of 2k + 1.
07
Derive contradiction
Since we assumed m and n are odd and m n turned out to be odd, this contradicts the given condition that m n is even.
08
Conclude the proof
Therefore, our assumption that both m and n are odd is incorrect. Hence, at least one of them must be even.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Even and Odd Integers
In mathematics, integers can be broadly classified into two categories: even and odd.
An integer is called even if it is divisible by 2 without a remainder. For example, 2, 4, 6, and 8 are all even numbers because they can be written in the form of 2k, where k is an integer.
On the other hand, an integer is called odd if it is not divisible by 2 without a remainder. Odd integers can be expressed as 2k + 1, where k is an integer. For example, 1, 3, 5, and 7 are all odd numbers.
Understanding whether an integer is even or odd is crucial for solving many mathematical problems, including proofs like the one in the exercise. Knowing the forms 2k and 2k + 1 helps in representing integers algebraically, making it easier to manipulate and understand their properties.
An integer is called even if it is divisible by 2 without a remainder. For example, 2, 4, 6, and 8 are all even numbers because they can be written in the form of 2k, where k is an integer.
On the other hand, an integer is called odd if it is not divisible by 2 without a remainder. Odd integers can be expressed as 2k + 1, where k is an integer. For example, 1, 3, 5, and 7 are all odd numbers.
Understanding whether an integer is even or odd is crucial for solving many mathematical problems, including proofs like the one in the exercise. Knowing the forms 2k and 2k + 1 helps in representing integers algebraically, making it easier to manipulate and understand their properties.
Definition of Parity
The term 'parity' in mathematics refers to whether an integer is even or odd. This concept is vital in number theory and plays a key role in various mathematical proofs and problems.
When we talk about the parity of an integer, we are essentially discussing its classification as either even or odd.
In the context of proofs, understanding parity helps us quickly identify properties of numbers and apply logical reasoning. For example, in the given exercise, parity helps determine the nature of products of integers.
Knowing that the product of two integers gives a result with specific parity can often lead to deeper insights or contradictions, which are useful in proof techniques like proof by contradiction.
When we talk about the parity of an integer, we are essentially discussing its classification as either even or odd.
- If an integer is even, it has an even parity.
- If an integer is odd, it has an odd parity.
In the context of proofs, understanding parity helps us quickly identify properties of numbers and apply logical reasoning. For example, in the given exercise, parity helps determine the nature of products of integers.
Knowing that the product of two integers gives a result with specific parity can often lead to deeper insights or contradictions, which are useful in proof techniques like proof by contradiction.
Integer Multiplication
Multiplying integers follows certain rules, especially when considering their parity.
When we multiply two integers, the result can be classified based on the parity of the integers involved:
These rules stem from how we represent even and odd numbers algebraically.
For instance, when multiplying two odd numbers, using their forms 2a + 1 and 2b + 1, you get (2a + 1)(2b + 1) = 4ab + 2a + 2b + 1 = 2(2ab + a + b) + 1.
This result is always odd since it follows the form 2k + 1.
By understanding these basic multiplication rules, the given proof by contradiction shows that if the product of two integers is even, at least one of them must be even.
This follows logically because if both integers were odd, their product would be odd too, which contradicts the initial condition that the product is even.
When we multiply two integers, the result can be classified based on the parity of the integers involved:
- Even 脳 Even = Even (e.g., 2脳4=8).
- Even 脳 Odd = Even (e.g., 2脳3=6).
- Odd 脳 Odd = Odd (e.g., 3脳5=15).
These rules stem from how we represent even and odd numbers algebraically.
For instance, when multiplying two odd numbers, using their forms 2a + 1 and 2b + 1, you get (2a + 1)(2b + 1) = 4ab + 2a + 2b + 1 = 2(2ab + a + b) + 1.
This result is always odd since it follows the form 2k + 1.
By understanding these basic multiplication rules, the given proof by contradiction shows that if the product of two integers is even, at least one of them must be even.
This follows logically because if both integers were odd, their product would be odd too, which contradicts the initial condition that the product is even.
