Week 4 – SS1 First Term General Mathematics Notes

 WEEK 4    
MODULAR ARITHMETIC
Modular arithmetic is a branch of Mathematics use to predict the outcomes of cyclic events such as days of the week, marketdays, months of the year, time etc.
    RULES OF MODULAR ARITHMETIC

• The modulo value must be greater than the number worked upon.
• When using cyclic pattern in adding numbers, you must count clock wise direction.
• In subtraction of numbers , you must count anti -clock wise direction

EXAMPLE:
The shorter hand of a clock points to 5 on a clock face. What number does it point to after 30 hours?
Solution
30 hours after = 11 o” clock.

1. Find the following numbers in their simplest form in modulo 4
2. 15 b. 102

   Solution

15(mod4) = 15÷4
3 remainder 3, therefore 3 the remainder is taken as 3 mod 4
102mod4 = 102 ÷4 = 25 remainder 2
102 (mod 4) = 2 mod 4
ADDITION OF MODULO ARITHMETIC
EXAMPLE
Find the following modulo addition
:a.42 28 (mod 8) b. 54 25 (mod 5)
Solution

1. 42 + 28 = 70 mod 8

   70 mod 8 = 6 mod 8

2. 54 + 25 = 79 mod 5

   79 mod 5 = 4 mod 5
   SUBTRACTION OF MODULO NUMBERS

Find the simplest form of the following in their giving moduli.

1. -5 mod 6 b. -17 mod 10 c. -75 mod 7

   SOLUTION

2. -5 mod 6 = -6×1+1 = 1 mod 6 the value added to the negative number to give the require result becomes your result.
3. -17 mod 10 = -10 x 2 + 3 = 3mod 10.
4. -75 mod 7 = -7 x 11+2 =2 mod 7.

MULTIPLICATION OF MODULO NUMBERS
Evaluate the following in their moduli.

1. 16 7 mod 5 b. 21 65 mod 4

   SOLUTION

A 16 X 7 = 117 mod 5, which is 2 mod 5

1. 21 x 65 = 1365 mod 4 = 1mod 4

   EQUATION OF MODULO

Solve the following equations in their giving moduli

1. 3x =5mod 7 b. 2x + 3 =1 mod 6 c.

   SOLUTION

   1. 3x =5+7 ,. 3x =12

      X = 4 mod 7

   2. 2x +3 =1+6 , 2x +3 =7

      2x 7 -5, then 2x = 4
      X =2 mod 6
      ASSESSMENT: Solve the following questions;

      1. Use the cycle number in modulo 6 to simplify the following.
         1. 2 + 10 (ii). 5 + 5 (iii) 15 + 37 (iv) 2 – 9 (v) 0 – 22
      2. A toy car starts at a point 0 and runs around a circular track of 2 meters. How far is the car from its starting point along the track when it has gone :
      1. 6m (b) 15m (c) 21m (d) 87m
         1. Find the following numbers in their simplest form in modulo 4:

            (i). 62(ii). 102 (iii) -56 (iv) -78 (v) -202

         2. Solve the following equations in the set of positive integers of each modular arithmetic:
            1. 3X + 4 = 7 mod 8 ii. 4X – 3 = 6 mod 7 iii. – 2X + 2 = 0 mod 5 IV. = 2 in (a). Mod 5 (b). Mod 6 (c). mod 9

                

 MORAL OBJECTIVE: EXODUS 15:25and he cried unto the Lord and there he proved them.