## How many integers must you pick in order to be sure that at least two of them have the same remainder when divided by 15?

Question

How many integers must you pick in order to be sure that at least two of them have the same remainder when divided by 15?

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4 weeks 2022-12-24T07:06:31+00:00 1 Answer 0 views 0

## Answer ( 1 )

1. Because of this, we need to choose 16 integers in such a way that we can ensure that when they are divided by 15, at least two of them will provide the same residual.
This is further explained below.

### How many integers must you pick in order to be sure that at least two of them have the same remainder when divided by 15?

Generally,  If you divide a number by 15, the result is a fraction. There are 15 remainders that might occur, and they are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, and 14.
That is to say, if we choose one number and divide it by 15, there is a chance that the remainder may be any of the 15 (0 reminders 14). Therefore, if we choose 15 numbers, we will have the opportunity to pick 15 unique reminders, ranging from 0 to 14.
In conclusion, if we select one more integer, that is we pick 16 integers, then at least two of them would have the same reminder since there are only 15 reminders and 16>15, and by the pigeonhole principle.
Because of this, we need to choose 16 numbers such that we can guarantee that at least two of them will provide the same remainder when they are divided by 15.
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