given any seven positive integers, show that at least two must exist whose sum or difference is divisible by 10.
To show that at least two integers exist whose sum or difference is divisible by 10, we can use the Pigeonhole Principle.
The Pigeonhole Principle states that if you have n+1 objects (pigeons) and distribute them into n boxes (pigeonholes), then at least one box must contain more than one object.
In this case, we can think of the integers as the objects and the possible remainders when dividing by 10 as the pigeonholes.
Since there are 7 integers and 10 possible remainders when dividing by 10 (0, 1, 2, ..., 9), we have more objects (7 integers) than pigeonholes (10 possible remainders).
By the Pigeonhole Principle, at least one remainder must contain more than one integer, meaning that there are at least two integers with the same remainder when divided by 10.
Now, consider two cases:
Case 1: Two integers have the same remainder when divided by 10:
In this case, the difference of these two integers will be divisible by 10. For example, if we have integers 12 and 22, their difference 22 - 12 = 10 is divisible by 10.
Case 2: Two integers have remainders that add up to 10:
In this case, the sum of these two integers will be divisible by 10. For example, if we have integers 17 and 3, their sum 17 + 3 = 20 is divisible by 10.
In either case, we have found two integers whose sum or difference is divisible by 10. Therefore, at least two integers must exist with this property, as required.