To show that among any group of five integers, there are two with the same remainder when divided by 4, we can use the pigeonhole principle.

Solution

1. Remainders when divided by 4:
   When an integer is divided by 4, the remainder can be one of the four values: $0, 1, 2,$ or $3$. Therefore, there are only four possible remainders (0, 1, 2, and 3) for any integer when divided by 4.

2. Five integers and the pigeonhole principle:
   If we select five integers, we have more integers (5) than possible remainder values (4). According to the pigeonhole principle, if we have more items (in this case, five integers) than categories (in this case, four possible remainders), then at least two of the items must fall into the same category.

3. Applying the principle:
   In our case, with five integers, at least two of them must share the same remainder when divided by 4. This is because there are only four possible remainder categories (0, 1, 2, and 3), but we have five integers.

Therefore, we conclude that among any five integers, there must be at least two integers with the same remainder when divided by 4.

Would you like more details or have any questions?

Here are five related questions to deepen your understanding:

1. How would this principle apply if you had six integers and three possible remainders?
2. Can the pigeonhole principle be applied to find two numbers with the same remainder among seven integers divided by 6?
3. How would this solution change if the integers were restricted to be consecutive?
4. What if we wanted to find a pair of integers with the same remainder when divided by 5 instead?
5. How can this principle be extended to groups with other integer divisions (e.g., groups of 7 with division by 3)?

Tip: The pigeonhole principle is a powerful tool in combinatorics, often used in problems involving distribution into limited categories or "pigeonholes."