Find the remainder when (19)^92 is divided by 92.

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  1. Interesting. Do you know how to get started on this problem?

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  2. Chinese Remainder theorem (along with other results). First note 92= 4 Γ— 23 with gcd
    (4,23) =1. Let us call N= 1992. We will compute, N(mod 4) and N (mod 23) and then use CRT to
    compute N (mod 92).
    First, N (mod 4) = (19)
    92( π‘šπ‘œπ‘‘ 4) = (βˆ’1)
    92(π‘šπ‘œπ‘‘ 4) = 1
    and 𝑁(π‘šπ‘œπ‘‘ 23) = 194
    22 (π‘šπ‘œπ‘‘ 23)]
    (π‘šπ‘œπ‘‘ 23) = (βˆ’4)
    (π‘šπ‘œπ‘‘ 23) = (16)
    (π‘šπ‘œπ‘‘ 23) = (βˆ’7)
    (π‘šπ‘œπ‘‘ 23) = 49
    (π‘šπ‘œπ‘‘ 23) = 3.
    Note in the above we have used Fermat’s Little Theorem. Now, If you know CRT, you can
    directly say 𝑁( π‘šπ‘œπ‘‘ 92) = 49.
    If not, you can compute it. One way to do it is write down two lists of numbers (one for each
    relation) and pick out the first common number

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