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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:

\(13 \times 40 \equiv 1 \times 4 (mod \ 6)\)

\(520 \equiv 4 (mod \ 6)\)