modular arithmetic

MODULAR ARITHMETIC

INTRODUCTION

Definition : MODULAR CONGRUENCE

Let n be an integer greater than 1 . The notation a ? b (mod n (1 will indicate that a and b are integers such that a - b is divisible by n . Statement (1 ) is called a modular congruence or simply a congruence and is read : `a is congruent to b , modulo n ' The integer is called the modulus of the congruence (Keesee , 1965

Illustration

13 ? 8 mod 5 is a congruence . When 13 is divided by 5 it yields a remainder equal to 3

. When 8 is divided by 5 , it yields a remainder of 3 . When 8 is subtracted from 13 the difference is 5 and 5 mod 5 0 which means that if 5 is divided by 5 , it yields a zero remainder

ADDITION OF MODULAR CONGRUENCES

Addition : If a1 ? a2 (mod n ) and b1 ? b2 (mod n ) then a1 b1 a2 b2 (mod n

Where a1 , a2 , b1 , b2 are elements of integers and a2 b2 (mod n must be reduced to integers between 0 to n - 1 by dividing a2 b2 by n . The remainder will be the final answer . If a2 b2 -1 5 4 ? -57 mod 5

-42 ? -67 mod 5

Therefore : -92 -42 ? -57 -67 mod 5 -134 -124 mod 5

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6-4 and -4 mod 5 1 mod 5 since

0 1 4 because (1 4 ) divided by 5 gives a 0 remainder

Or -124 120 -4 ( -4 5 1

MULTIPLICATION

If a1 ? a2 (mod n ) and b1 ? b2 (mod n

Then a1 b1 ? a2 b2 (mod n ) where a1 , a2 , b1 , b2 are elements of integers and a2b2 (mod n ) must be reduced to integers between 0 to n - 1 by dividing a2 b2 by n . The remainder will be the final answer

Illustration : Using positive integers (Ex . 1 ) Given the congruences

7 ? 12 mod 5 and 21 ? 26 mod 5

Therefore (7 (21 (12 (26 ) mod 5 147 ? 312 mod 5

simlifying furtherly 147 312 mod 5 2 mod 5 since when 312 is divided by 5 it yields a remainder 2

Ex . 2 ) 18 33 mod 5 and 42 32 mod 5

Therefore (18 (42 (33 (32 ) mod 5 ? 1056 mod 5

simplifying furtherly : 756 1056 mod 5 1 mod 5

since when 1056 is divided by 5 it yield a remainder 1

Example using Negative Integers (Ex . 1 ) Given -19 -24 mod 5 and -28 -33 mod 5

Therefore (-19 (-28 (-24 (-33 ) mod 5 ? 792 mod 5

simplifying furtherly : 532 792 mod 5 2mod 5

Since when 792 is divided b 5 it yields a remainder 2 (Ex . 2 ) Given : -18 -17 mod 5 and -23 -132 mod 5

Therefore (-18 (-23 (-17 (-132 ) mod 5

414 2244 mod 5

simplifying furtherly : 414 2244 mod 5 4 mod 5

since 2244 divided by 5 yields a remainder 4

References : Keesee , John W (1965

Elementary Abstract Algebra . U .S .A...