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.

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

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.

Questions & Answers

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.