I have the following problem:
"Let P(n) denote “If any n even integers are selected, then there must be a pair who share the same remainder under integer division by 12.”
Let n* denote the smallest value of n for which P(n) is true. What is n*? For your stated value of n*, prove that P(n*) is true and that P(n*−1) is false.[Note:This proves that your stated value of n* is the smallest value of n for which P(n) is true.]"
I am not sure how to apply the pigeonhole principle in this problem so as to prove it. Can someone help me or guide me in the right direction? We did very brief examples in class and nothing similar to this.
I noticed the possible remainders will be 12 from 0-11 , so i guess these are the pigeonholes but it is unclear how i can proceed.