Question

Prove that in any group of five integers, at least two have the same value under the $(mod4)$ operation.

Short answer

By the pigeonhole principle, at least two integers must have the same remainder mod 4.

Step-by-step solution

Understand the Problem

We need to prove that among any five integers, at least two will have the same remainder when divided by 4, also known as their value mod 4. This means looking at the possible remainders after division by 4.

List Possible Modulo Remainders

When any integer is divided by 4, the remainder can be 0, 1, 2, or 3. These four possibilities cover all outcomes for the mod 4 operation.

Utilize the Pigeonhole Principle

The pigeonhole principle states that if you have more items than containers, at least one container must hold more than one item. Here, our items are the five integers and our four containers are the possible remainders (0, 1, 2, 3).

Apply the Principle to the Problem

With five integers and only four possible remainders after division by 4, there's no way to place each integer in a separate "container" without some "container" being used more than once. This guarantees that at least two integers will have the same remainder, i.e., the same value mod 4.

Key concepts

Pigeonhole Principle

The Pigeonhole Principle is a simple yet powerful concept used in combinatorics and probability. It helps us to make conclusions about the arrangement or distribution of objects. The heart of this principle is that if you try to place more items into fewer containers, then at least one container must contain more than one item. This intuitive idea can be visualized using everyday scenarios, such as trying to fit five papers into four folders. Clearly, at least one folder must contain more than one paper.

In the context of our exercise, consider our items to be the remainders when five integers undergo a modulo operation with 4. The containers, in this case, are the potential remainders: 0, 1, 2, and 3. Since there are five integers but only four remainders under modulo 4, the Pigeonhole Principle tells us that at least two of these integers will have the same remainder. This principle is a fundamental reasoning tool in mathematics, often used in proofs and problem-solving to simplify seemingly complex distribution problems.

Integer division

Integer division is the process of dividing one integer by another and finding the quotient and remainder. For instance, when you perform integer division of 7 by 4, you are essentially determining how many times 4 can fit into 7 without exceeding it. For 7 divided by 4, the quotient is 1, and the remainder is 3, since 4 fits into 7 once, leaving 3 as the leftover.

The formula for integer division can be expressed as: $$a=bq+r$$ where:

• $a$ is the dividend (the number being divided),
• $b$ is the divisor (the number by which $a$ is divided),
• $q$ is the quotient (the integer result of the division),
• and $r$ is the remainder (what's left over).

The remainder $r$ follows the condition $0\le r<b$. Understanding integer division is essential for working with modulo operations, as it provides the structure to calculate and interpret remainders properly.

Remainder concept

Remainders are what is left after performing division. They are key to many mathematical operations, especially in the field of modular arithmetic. When we divide an integer by another integer, the remainder is the part of the dividend that isn't fully divisible by the divisor. For example, when dividing 10 by 4, the quotient is 2 and the remainder is 2, since 4 fits twice into 10 with 2 left over.

In a modulo operation, the remainder plays a central role. The notation $amodb$ represents the remainder of dividing $a$ by $b$. For $10mod4$, the result is 2. It highlights how remainders cycle through a set pattern. When dealing with a particular divisor, say 4, possible remainders are limited to a predictable cycle of 0 to 3.

This cyclical nature is integral to many mathematical areas and allows us to predict outcomes and simplify problems. Like in the given problem, understanding remainders reveals that two different numbers can behave equivalently in modular arithmetic, which is why at least two integers in any group of five will have the same remainder when divided by 4.