Question

Prove that if six integers are chosen at random, then at least two of them will have the same remainder when divided by 5 .

Short answer

By the Pigeonhole Principle, because there are 6 numbers and only 5 possible remainders when an integer is divided by 5, at least two of the 6 random integers must have the same remainder when divided by 5.

Step-by-step solution

Understand the problem

We are given that 6 integers are chosen at random and we are asked to prove that at least two of them will have the same remainder when divided by 5.

Apply the Pigeonhole Principle

The Pigeonhole Principle states that if n items are put into m containers, with n > m > 0, then at least one container must contain more than one item. In this problem, 'items' are the integers and 'containers' are the possible remainders when an integer is divided by 5. There are only 5 possible remainders: 0, 1, 2, 3, and 4. If we choose 6 numbers, 'n' becomes 6 which is more than the possible remainders 'm', since 'm' is 5.

Draw a conclusion

By the Pigeonhole Principle, because we have 6 numbers (n) and only 5 possible remainders (m), at least one remainder must be the same for at least two of the six integers, that is, at least two numbers must give the same remainder when divided by 5.