UPSKILL MATH PLUS

Learn Mathematics through our AI based learning portal with the support of our Academic Experts!

Learn moreIn 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:

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

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