Question

Given 3 integers, must there be two with the same remainder when divided by 3?

A) Yes
B) No

Answer 1

When dividing 3 integers by 3, the Pigeonhole Principle ensures that at least two must have the same remainder, as there are only three possible remainders: 0, 1, or 2.

Step-by-step explanation:

The question asks whether given 3 integers, there must be two with the same remainder when divided by 3. According to the Pigeonhole Principle, if three integers are each divided by 3, there are only three possible remainders: 0, 1, or 2. Since we have three integers and only three possible remainders, at least two of the integers must share the same remainder, as there are not enough 'pigeonholes' (remainders) for each 'pigeon' (integer) to have a unique place. Therefore, the answer to the question is A) Yes.