Modular Addition, Subtraction and Multiplication - SS1 Mathematics Lesson Note

The symbols \(\oplus ,\ \ \ominus ,\ \ \)and \(\otimes\) denote modular addition, subtraction, and multiplication respectively.

Example: Evaluate the following: (i) \(4 \oplus 9 \ominus 7(mod\ 2)\) (ii) \(72\ \ominus 81 \circleddash \ 13\ (mod\ 7)\) (iii) \(35\ \otimes \ 10\ (mod\ 6)\)

 

Solution

(i) \(4 \oplus 9 \ominus 7(mod\ 2)\ \ \ \ = 13 \ominus 7(mod\ 2)\ \ \ \ = \ 5\ (mod\ 2)\ \ \ \ = \ \frac{5}{2} = 2\ remainder\ \mathbf{1}\)

Thus, \(4 \oplus 9 \ominus 7(mod\ 2) \equiv \mathbf{1}\)

(ii) \(72\ \ominus 81 \circleddash \ 13\ (mod\ 7)\ \ \ = \ - 9 \circleddash \ 13\ (mod\ 7)\ \ \ = - 22\ (mod\ 7)\)

\(- 22 = x\ (mod\ 7)\)

\(- 22 = q.7 + \mathbf{x}\)

\(- 22 = - 4(7) + \mathbf{x}\)

\(- 22 = \ - 28 + \mathbf{x}\)

\(- 22 + 28 = \mathbf{x}\)

\(\mathbf{x =}6\)

Thus, \(72\ \ominus 81 \circleddash \ 13\ (mod\ 7) \equiv \mathbf{6}\)

(iii) \(35\ \otimes \ 10\ (mod\ 6)\ \ \ \ = 350\ (mod\ 6)\)

\(\frac{350}{6} = 58\ remainder\ \mathbf{2}\)

\(35\ \otimes \ 10\ (mod\ 6)\mathbf{\equiv 2}\)

 

Example: What is the time \(16\ hours\) after \(6\ o’clock\) (\(6:00\ am\))?

Solution

Assuming a 12-hour clock, that is modulo 12

\(16\ \oplus 6\ (mod\ 12)\ \ \ \ \ = 22\ (mod\ 12)\ \ \ \ \ \ = \ \frac{22}{12}\ = 1\ remainder\ \mathbf{10}\)

Thus, \(16\ \oplus 6\ (mod\ 12)\ \equiv \mathbf{10}\), meaning the time will be 10 o’clock or 10:00 pm that same day.

Assuming a 24-hour clock, that is modulo 24

\(16\ \oplus 6\ (mod\ 24)\ \ \ \ \ = 22\ (mod\ 24)\ \ \ \ \ \ = \ \frac{22}{24}\ = 0\ remainder\ \mathbf{22}\)

Thus, \(16\ \oplus 6\ (mod\ 24)\ \equiv \mathbf{22}\), meaning the time will be 22:00 hours or 10:00 pm that same day.