Prove that Every Integer is Even or Odd | Summary and Q&A

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April 29, 2022
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The Math Sorcerer
Prove that Every Integer is Even or Odd

TL;DR

Every integer can be written as either 2k or 2k+1, where k is an integer.

Key Insights

• 🔨 The division algorithm is a powerful tool in number theory.
• #️⃣ Every integer can be expressed as either an even number or an odd number.
• 🦕 The remainder when dividing by 2 determines whether an integer is even or odd.

Transcript

hello in this problem we're going to prove that every integer is either even or odd so by even we mean that it has the form say 2k where k is some integer and by odd we mean it has the form say 2k plus 1 where k is an integer this is almost an immediate consequence of something called the division algorithm so before we go through the proof i'm jus... Read More

Q: What is the division algorithm?

The division algorithm states that for any positive integers a and b, there exist unique integers q and r such that a = bq + r, where r is the remainder.

Q: How can the division algorithm be applied to prove that every integer is either even or odd?

By choosing b = 2 and applying the division algorithm, it can be shown that every integer n can be written as either 2k or 2k+1, where k is an integer.

Q: What does it mean for an integer to be even?

An integer is even if it can be written in the form 2k, where k is an integer.

Q: What does it mean for an integer to be odd?

An integer is odd if it can be written in the form 2k+1, where k is an integer.

Summary & Key Takeaways

• The content introduces the division algorithm, which states that for any positive integers a and b, there exist unique integers q and r such that a = bq + r, where r is the remainder.

• By applying the division algorithm with b = 2, it can be proven that every integer n can be written as either 2k or 2k+1, where k is an integer.

• If the remainder r is 0, then n is even, and if the remainder is 1, then n is odd.