Multiple of 9
Number Theory
Level
2
Consider a set consisting of 10 natural numbers. If we calculate all the differences between these numbers, at least one of these differences is a multiple of 9?
False
True
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1 solution
Let's look at those numbers modulo 9 .
In modulo 9, there are 9 equivalence classes (0,1,2,...,7,8) and since we have 10 numbers, by the pigeinhole principle , there must be at least two numbers that are in the same equivalence class. This directly implies that their difference is a multiple of 9.