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Using pigeonhole principle prove that in any group of 5 integers, not necessarily consecutive, there are...

Using pigeonhole principle prove that in any group of 5 integers, not necessarily consecutive, there are 2 with same remainder when divided by 9

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Answer #1

No, it cannot happen at all.

Since the numbers are divided by 9 and it has more possible number of remainders than 5.

There are nine possible remainders when an integer is divided by 9: 0, 1, 2, 3, 4, 5, 6, 7 or 8 (these are pigeonholes). But when a group of 5 integers is divided by 9 (no doubt that the remainder will be from the above numbers) but not necessarily there are 2 with same remainder as .

it can happen when the group of 5 integers, not necessarily consecutive, when divided by any number less than 5 obviously greater than 1.

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