Modular Arithmetic – Addition, Subtraction, Multiplication operations and Applications

CONTENT

• Concept of Modular Arithmetic
• Addition, Subtraction and Multiplication Operations in Module Arithmetic
• Application to daily life.

Modular Arithmetic

In the previous section, we discovered a new kind of arithmetic, where we add positive integers by roating in number cycle. This arithmetic is called modular arithmetic. In our example, we ignored multiples of 4 and concentrated on the remainders. In this case we say that the modulus is 4

For example,

5 = 1 (mod 4)

Where mod 4 means with modulus 4 or modulo 4.

Note that 9÷ 4 = 2, remainder 1

And 45 ÷ 4 = 11 remainder 1

We say that 9 and 45 are equal modulo 4,

i.e. 9 = 45 = 1 (mod 4)

Example 1

Reduce 55 to its simplest form:

Modulo 3

Modulo 4

Modulo 5

Modulo 6

55 ÷ 3 = 18, remainder 1

55 = 1 (mod 3)

55 ÷ 4 = 13, remainder 3

55 = 3 (mod 4)

55 ÷ 5 = 11, remainder 0

55 = 0 (mod 5)

55 ÷ 6 = 9, remainder 1

55 = 1 (mod 6)

EVALUATION

Write down the names of four markets in your locality which are held in rotation over 4* days.

Addition, Subtraction and Multiplication Operations in Module Arithmetic

Addition and Subtraction

The table below shows an addition table (mod 4) in which numbers 0, 1, 2 and 3 are added to themselves.

READING ASSIGNMENT

New General Mathematics for SS 1 Page 239 ex. 20c 1 – 10

WEEKEND ASSIGNMENT

Find the simplest form of the following in the given moduli.

1. -75 (mod 7)A. 4    B. 2    C. 5      D. 7
   1. -56 (mod 13)A. 10     B. 5    C. 9     D. 12