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Math Help - largest integer which gives same modulus result

  1. #1
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    largest integer which gives same modulus result

    Hi, if we have a set of positive integers then how can one find the largest positive integer say M such that we get same result on taking modulus operation of each integer in the set with M? E.g if the given set is 12,16,18,22 then then M would be 2 , as 12%2 == 16%2 == 18%2==22%2 . Do we need to find the LCM of the numbers in set ? Thanks.
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  2. #2
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    Re: largest integer which gives same modulus result

    Suppose we have two relatively prime numbers, say 18 and 31. LCM has nothing to do with it, but GCF(18, 31) = 1. However, M is not 1. 18 and 31 are congruent mod 13 so M = 13.

    Turns out, if we have n numbers, a_1, a_2, ..., a_n, M should be equal to the GCF of all of the pairwise differences a_i - a_j. This is because if M is the largest modulus such that all the a_i are congruent, then a_i - a_j \equiv 0 (\mod M) for all i,j. Therefore M must be a factor of all the pairwise differences, and the largest such M is the GCF of these differences.
    Thanks from pranay
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