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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:

$$13 + 40 \equiv 1 + 4 (mod \ 6)$$

$$53 \equiv 5 (mod \ 6)$$

(ii) Subtraction:

$$13 - 40 \equiv 1 - 4 (mod \ 6)$$

$$-27 \equiv -3 (mod \ 6)$$

(iii) Multiplication:

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

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