Question

# Given a set of seven distinct positive integers, prove that there is a pair whose sum or whose difference is a multiple of 10. you may use the fact that if the one digit is a 0 it is divisible by 10. hint: create six categories of integers based on their ones digit. I cant figure out the six categories.

Answer

We can group the integers based on their last digit. If none of the ten pairs whose difference is divisible by 10 are in the set, there must be at least two integers with the same last digit, and these two integers form a pair whose difference is divisible by 10. If any of the ten pairs whose difference is divisible by 10 are in the set, then the statement is obviously true.

- Q: How can we create the six categories?
- Q: What is the justification behind creating these categories?
- A: If none of the ten pairs whose difference is divisible by 10 are in the set, then there are at most six possible last digits for the seven integers. By the pigeonhole principle, there must be at least two integers with the same last digit. These two integers form a pair whose difference is divisible by 10. If any of the ten pairs whose difference is divisible by 10 are in the set, then the statement is obviously true.