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mod arithmetic

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mod arithmetic a mod m is the remainder of a divided by m a mod m is the integer r such that a = qm + r and 0 <= r < m again, r is positive Examples 17 mod 3 = 2 17 mod 12 = 5 (5 o’clock) -17 mod 3 = 1

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congruences a is congruent to b modulo m if m divides a - b

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**a is congruent to b mod m if and only if the remainder of a**

divided by m is equal to the remainder of b divided by m. proof

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**If a is congruent to b mod m and c is congruent to d mod m**

then a+c is congruent to b+d mod m proof

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**If a is congruent to b mod m and c is congruent to d mod m**

then ac is congruent to bd mod m proof

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Mod arithmetic examples -133 mod 9 = 2 (but in Claire?) list 5 numbers that are congruent to 4 modulo 12 hash function h(k) = k mod 101 h( ) h( ) h( ) h( )

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Number-Theoretic Algorithms

Number-Theoretic Algorithms

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