Theory:

In this topic, we shall learn how to perform arithmetic operations in modulo arithmetic.
 
Consider the following theorems.
Theorem I: \(a\), \(b\), \(c\) and \(d\) are integers and \(m\) is a positive integer such that \(a \equiv b (mod \ m)\) and \(c \equiv d (mod \ m)\) then
 
(i) \((a + c) \equiv (b + d) (mod \ m)\)
 
(ii) \((a - c) \equiv (b - d) (mod \ m)\)
 
(iii) \((a \times c) \equiv (b \times d) (mod \ m)\)
 
Theorem II: If \(a \equiv b (mod \ m)\) then
 
(i) \(ac \equiv bc (mod \ m)\)
 
(ii) \(a \pm c \equiv b \pm c (mod \ m)\) for any integer \(c\).
Example:
If \(13 \equiv 1 (mod \ 6)\) and \(40 \equiv 4 (mod \ 6)\), then apply theorem I and find the values using addition, subtraction and multiplication of the given modulo.
 
Solution:
 
(i) Addition:
 
\(13 + 40 \equiv 1 + 4 (mod \ 6)\)
 
\(53 \equiv 5 (mod \ 6)\)
 
(ii) Subtraction:
 
\(13 - 40 \equiv 1 - 4 (mod \ 6)\)
 
\(-27 \equiv -3 (mod \ 6)\)
 
(iii) Multiplication:
 
\(13 \times 40 \equiv 1 \times 4 (mod \ 6)\)
 
\(520 \equiv 4 (mod \ 6)\)