Modular Match


Ishfaq has randomly chosen $4$ numbers. Prove that it is possible to choose two of those numbers such that their difference is divisible by $3$.


Source: BdMO 2017 National Junior P6


Proof Based Problems  


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Solution

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Let the numbers be $a,b,c,d$.

Modulo $3$, there are $3$ possible values which are $0, 1, 2$.

So, using Pigeon Hole Principle, at least $2$ of the numbers are equal modulo $3$.


Without loss of generality, assume that $a\equiv b \pmod{3}$.

So, $|a-b|\equiv 0\pmod 3$

So, the difference of $2$ numbers is divisible by $3$.

This is a proof based problem added for learning purposes and does not accept submissions.

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