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Discrete Math: 8) Prove that among any set of seven (not necessarily consecutive) integers, there are...

Discrete Math:

8) Prove that among any set of seven (not necessarily consecutive) integers, there are at least two with the same remainder when divided by 6.

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Answer #1

When divided by 6 there are only 6 possible remainders (0, 1, 2, 3, 4, 5)

But there are 7 numbers So, there will be exactly seven remainders from those 6 possible remainders only i.e. (0, 1, 2, 3, 4, 5)

By pigeon hole principal 7 = 6(1) +1 {corresponding to worst case when we have all 6 possibilities from 0 to 5)

This means, there are at least two with the same remainder when divided by 6.

Hence Proved

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