Given $7$ integers such that no two have the same remainder on division by $20$, prove that we can choose $4$ of those $7$ numbers, $\{a,b,c, d \}$, such that $a+b-c-d$ is divisible by $20$.
2026-04-01 21:08:45.1775077725
pigeonhole principle proof for division by $20$
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How many different pairs of integers can you make from 7 integers (Where we count (m,n) as being the same as (n,m)) ?
Can you show that, from these pairs of integers, at least 2 pairs (call them (a,b) and (c,d)) have the same sum modulo 20 ?
What does that tell you about a+b-c-d ?