There are 45 candidates appear in an examination. prove that there are at-least two candidates in class whose roll numbers differ by a multiple of 44.
How can I prove this using pigeonhole principle?
2026-04-01 09:52:16.1775037136
Solve using Pigeonhole principle
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Consider the possible remainders mod $44$ as the boxes. There are $44$ of these.
Now there are $45$ roll numbers in total. Place each in its box corresponding to its remainder mod $44$.
The pidgeonhole principle says that there will be at least two roll numbers $a,b$ such that they lie in the same box, i.e. $a\equiv b \bmod 44$.
But then $44|(a-b)$.