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MODULAR ARITHMETIC

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By the de nition of a \remainder," we can write a = im + r 1, where r 1 is the remainder under division by m and satis es 0 r 1 m 1. Similarly, b = jm + r 2 with 0 r 2 m 1. Then if m divides a b, this means that m divides im + r 1 jm r 2 = m(i j) + r 1 r 2. Since m clearly divides m(i j), we get that m divides r 1 r 2. We know that r 1 and r

  Modular, Arithmetic, Remainder, Modular arithmetic

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