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.  

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