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Introduction
Chinese Remainder
Theorem
Discrete Mathematics
Andrei Bulatov
Discrete Mathematics – Chinese Remainder Theorem
342
Previous Lecture
Residues and arithmetic operations
Caesar cipher
Pseudorandom generators
Divisors of zero
Inverse
Discrete Mathematics – Chinese Remainder Theorem
343
Linear Congruences
A congruence of the form
ax
≡
b (mod m)
where
m
is a positive integer,
a
and
b
are integers, and
x
is a
variable,
is called a
linear congruence
.
We will solve linear congruences
If
a
is relatively prime with
m,
then it has the inverse
.
Then
⋅
ax
≡
⋅
b (mod m)
x
≡
⋅
b (mod m)
Find the inverse of 3 modulo 7
Solve the linear congruence
3x
≡
4 (mod m)
1
a

1
a

1
a

1
a

Discrete Mathematics – Chinese Remainder Theorem
344
The Chinese Remainder Theorem
A linear congruence is similar to a single linear equation. What
about systems of equations
(Sun Tzu’s puzzle, 400 – 460 BC):
“There are certain things whose number is unknown. When divided
by 3, the remainder is 2;
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This note was uploaded on 05/18/2010 for the course MACM MACM 101 taught by Professor Andreibulatov during the Spring '10 term at Simon Fraser.
 Spring '10
 AndreiBulatov

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