Lesson Notes By Weeks and Term - Senior Secondary School 1

SUBJECT: MATHEMATICS

CLASS:  SS 1

DATE:

TERM: 1st TERM

REFERENCE BOOKS

• New General Mathematics SSS 1     M.F. Macrae et al 

 
WEEK FOUR

TOPIC: Modular Arithmetic

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: 

1. Modulo 3
2. Modulo 4 
3. Modulo 5
4. Modulo 6

1. 55 3 = 18, remainder 1 

55 = 1 (mod 3)

1. 55 4 = 13, remainder 3

55 = 3 (mod 4)

1. 55 5 = 11, remainder 0

55 = 0 (mod 5)

1. 55 6 = 9, remainder 1 

55 = 1 (mod 6)

EVALUATION

1. 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. 

                        Second number

In the table, multiples of 4 are ignored and remainders are written down. For example 2 ⨁ 3 = 5 = 1 (mod 4) and 2 ⨁ 2 = 4 = 0 (mod 4.) note that we often use the symbol ⨁ to show addition in modular arithmetic.

Example 1 

Find a. 0 3 (mod 4),    b. 1 2 (mod 4)

1. Start at 0 and move in an anticlockwise direction three places. 

The result is 1.

Therefore, 0 3 = 1 (mod 4)

1. Start at 1 and move in an anticlockwise direction two places. The result is 3. 

Therefore, 1 2 = 3 (mod 4).

Second number

Notice the importance here of stating which number comes first, e.g. 2 1 1 2

Example 2 

Add 39 ⨁ 29 (mod 6)

Either 

39 ⨁ 29 = 68

    = (6 x 11 + 2)

    = 2 (mod 6)

Or, expressing both numbers in mod 6

39 ⨁ 29 = (6 x 6 + 3) + (6 x 4 + 5)

    = (3 + 5) (mod 6)

    = 8 (mod 6)

    =2 (mod 6)

Multiplication 

Example 1 

Evaluate the following, modulo 4,

1. 2 ⨂ 2         b. 3 ⨂ 2        c. 33 ⨂ 9

1. 2 ⨂ 2 = 4 (mod 4)
2. 3 ⨂ 2 = 4 + 2 = 2 (mod 4)
3. 33 ⨂ 9 = 297 = 4 x 74 + 1 = 1 (mod 4)

Or expressing both numbers in mod 4 

33 ⨂ 9 = 1 x 1 (mod 4)

        = 1 (mod 4)

Example 2 

Evaluate the following in the given moduli.

1. 16 ⨂ 7 (mod 5) b. 18 ⨂ 17 (mod 3)
2. 16 ⨂ 7 = 112

    = 22 ⨂ 5 + 2

    = 2 (mod 5)

or

16 = 15 + 1 = 1 (mod 5)

7 = 5 + 2 = 2 (mod 5)

16 ⨂ 7 = 1 ⨂ 2 (mod 5)

    = 2 (mod 5)

1. 18 ⨂ 7 (mod 3)

18 = 0 (mod 3)

17 = 2 (mod 3)

18 ⨂ 17 = 0 ⨂ 2 (mod 3)

        = 0 (mod 3)

In examples 1, 2, it can be seen that it is usually most convenient to convert the given numbers to their simplest form before calculation. 

EVALUATION 

1. Find the following numbers in their simplest form, modulo 4. 
   1. 15 
   2. 102
   3. 38
2. Find the values in the  moduli written beside them.
   1. 16 ⨂ 7 (mod 5)
   2. 80 ⨂ 29 (mod 7)
   3. 21 ⨂ 18 (mod 10)

GENERAL EVALUATION 

1. Complete the multiplication modulo 5

1. a. The shorter hand of a clock points a 10. What number did it point to 29 hours ago?

1. find the simplest positive form of -29 (mod 12)
2. Calculate 10 – 29 (mod 12)

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

Find the values in the moduli written beside them. 

1. 8 ⨂ 25 (mod 3) A. 2     B. 5     C. 9    D. 4
2. 27 ⨂ 4 (mod 7)A. 7     B. 5     C. 1    D. 3
3. 21 ⨂ 65 (mod 4)  A. 1    B. 9      C. 4     D. 8 

THEORY 

• Calculate the following 

1. 42 ⨁28 (mod 8)
2. 12 ⨁ 9 (mod 4)
1. Complete the multiplication modulo 6