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13 May, 17:22

Problem 4 (3 pts) : Let n be a positive integer. Show that among any group of n 1 (not necessarily consecutive) positive integers there are at least two with the same reminder when they are divided by n.

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  1. 13 May, 20:08
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    Answer:There are two integers in the group of n+1 integers with exactly the same remainder when they are divided by n.

    Explanation:

    Generally, if a number is divided by p (positive integer), then the possible remainders will be from 0 to p-1.

    Here, the possible remainders when an integer is divided by n are 0,1, ..., n-1

    so the number of possible remainders when an integer is divided by n is n.

    In this case, the number of objects is n+1 integers and the number of boxes (remainders) is n.

    p/k = (n+1) / n

    = 1 + (1/n)

    = 2

    Here, 0<1/n<1

    Add 1 on both sides to get the following

    0+1 < 1+1/n<1+1

    1<1+1/n<2

    so the value of p/k = 2 means that there is atleast one remainder which is same for two integers when they are divided by n

    There are therefore two integers in the group of n+1 integers with exactly the same remainder when they are divided by n.
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