Modulo operations — lesson. Mathematics State Board, Class 10.

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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:

$13 \equiv 1 (mod \ 6)$ 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:

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

$520 \equiv 4 (mod \ 6)$

Did you find an error?

4. Modulo operations