Operations in Modular Arithmetic (Addition, Subtraction, Multiplication and Equations.)

Addition and Subtraction in Modular Arithmetic:

To add and subtract in modular arithmetic, only add or subtract the numbers and find the remainder when the sum is divided by the giving modulus.

For example:

(1). Simplify:

a.       57 + 28 mod₆

b.      38 + 6 mod₇

c.       18 – 7 mod₄

(2). Simplify:

a.       -27 mod₈

b.      -31 mod₃

Solution:

(1).      

a.       (57 + 28) mod₆ ≡ (57 + 28) mod₆

 ≡ 85 mod₆

 ≡ 1 mod₆ [85 divided by 6 = 14 remainder 1]

Try the b). and c)., on your own.

(2).

a.       -27 mod₈ ≡ -(27) mod₈

     ≡ -3mod₈

     ≡ -3 + 8 mod₈

     ≡ 5 mod₈

Try the b)., on your own.

Multiplying in Modular Arithmetic:

To multiply in modular arithmetic, find the product and then use the modulus divide the result and find the remainder.

For example, simplify the following:

a.       15 × 7 mod₅

b.      13 × 9 mod₆

Solution:

a.       15 × 7 ≡ (15 × 7) mod₅

≡ (105) mod₅

≡ 0 mod₅

Try the b)., on your own.

Addition and Multiplication table in modular arithmetic:

Addition (+) and Multiplication (×) table can be constructed in any modulus. For example; construct a table for addition in mod₄ and use your table to find the following:

1.      2 (+) 3 mod₄

2.      2 (+) 3 (+) 3 mod₄

Solution:

Giving that mod₄ = {0, 1, 2, 3}

(+)	0	1	2	3
0	0	1	2	3
1	1	2	3	0
2	2	3	0	1
3	3	0	1	2

From the above table:

1.      2 (+) 3 mod₄ = 1 mod₄

2.      2 (+) 3 (+) 3 mod₄ = 0 mod₄

Example (2):

Construct addition and multiplication table for mod₅. Using your tables, find the truth set of the following:

a.       4 × (n + 4) = 3

b.      (n +1) × (n + 1) = 4.

Please try the above example on your own.

Equations in Modular Arithmetic:

You can solve equations in modular arithmetic. For example:

a.       If 4 × 3 ≡ x mod_6, find the value of x.

b.      If 2x + 1 ≡ 7 mod_8, find the value of x.

Solution:

a.       Giving that 4 × 3 ≡ x mod_6,

But 4 × 3 ≡ (4 × 3) mod₆

                ≡ (12) mod₆

                ≡ 0 mod₆

∴ the value of x is 0.

b.      2x + 1 ≡ 7

      2x ≡ 7 – 1

     2x  ≡ 6

        x ≡ 3

∴ the value of x is 3.

Try the following examples on your own.

a.       Find the value(s) of 3x + 4 ≡ 0 mod₅

b.      Find the solution set of x² + 1 ≡ 3 mod_7.  

You may also like these:

1.   Indices - New!

2.  Modular Arithmetic - An introduction - New!

3.   Negative Indices, Zero Indices and Exponents that are Fractions: - New!

4.  The Venn diagram – Two-Set Problem. - New!

For any further advice on your studies or any challenge on this post, please feel free to contact me at: dzaazo@gmail.com. I will appreciate it.

Modular Arithmetic - An introduction

Modular Arithmetic is a special base arithmetic in which only the remainder is reported. That is, it can be considered as remainder arithmetic.

For example, considering the number of days of the week on a circular dial or circular clock face as shown below.

 

If we take Sunday as the starting point (0), Monday (1) and so on clockwise around the dial, we can find any day we want. For example, starting at Sunday, what day will be in 18 days?

Solution:

When we start on Sunday, we will return to Sunday every seven days. So 18 days from Sunday will be 2 (7) + 4; this is Thursday.

We could also solve this problem by using the idea of remainder. That is dividing 18 by 7 then we consider the remainder.

 

Now the remainder (4) shows there is four days after Sunday and that is Thursday.

From the above the example, the number of days in the cycle is called its modulus and each day is called a cyclic variable. This type of circular arithmetic is called Modular Arithmetic.

NB: The modular arithmetic base 7 is written as mod₇, base 9 is written as mod₉ and so on. In general, in any mod_n arithmetic, the remainder will form a set with elements {0, 1, 2 ... n – 1}

So that if n = 7, mod₇ = {0, 1, 2, 3, 4, 5, 6}

                n = 10, mod₁₀ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and so on.

The equivalent sign (≡) is used in modular arithmetic.

Equivalent Values in Modular Arithmetic:

Two or more number are said to be equivalent if they give the same remainder when divided by a particular modulo. For numbers such as 3, 8, 13, 18, ... are equivalent in arithmetic modulo 5.

Example: simplify the following

1.      17mod₃

2.      28mod₉

3.      125mod₁₀

Solution:

1.      17mod₃ ≡ 2mod₃. This is because (17 divided by 3 = 5 remainder 2.)

2.      28mod₉ ≡ 1mod₉. This is because (28 divided by 9 = 3 remainder 1.)

3.      125mod₁₀ ≡ 5mod₁₀. This is because (125 divided by 10 = 12 remainder 5.)

You may also like these:
1.   Indices - New!
2.  Negative Indices, Zero Indices and Exponents that are Fractions: - New!
3.   Operations in Modular Arithmetic (Addition, Subtraction, Multiplication and Equations.) - New!
4.  The Venn diagram – Two-Set Problem. - New!

For any further advice on your studies or any challenge on this post, please feel free to contact me at: dzaazo@gmail.com. I will appreciate it.