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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.

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Presentation on theme: "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."— Presentation transcript:

1 mod arithmetic

2 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 oclock) -17 mod 3 = 1

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

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7 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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11 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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14 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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16 Mod arithmetic -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(104578690) h(432222187) h(372201919) h(501338753) examples


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