Lesson 6, Topic 1
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# Multiplication & Division in Modular Arithmetic

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Multiplication in Modular Arithmetic

Multiplication can be explained as repeated addition e.g. 4 ⊗ 3 (mod 5) i.e.

$$\scriptsize 3 \: + \: 3 \: + \: 3 \: + \: 3 = 12 \\ \scriptsize = 2(mod 5)$$

i.e.  $$\scriptsize 4 \: \bigotimes \: 3 = 2 (mod\:5)$$

The table below shows multiplication in modulo 5.

For example, 3 ⊗ 2 means 3 on the residues column against 2 on the residues row, the result is 1.

i.e.    3 ⊗ 2 (mod 5) = 1 (mod 5)

Division

As a matter of fact, division is the inverse of multiplication, thus we can use multiplication table to solve division problems.

For example, 3 ⨸ 4 = ? (mod 5)

Let 3 ⨸ 4 = a

Then 4a = 3  $$\scriptsize since \left (\; \normalsize \frac{3}{4} = \scriptsize a, \; then \; 4a = 3 \right)$$

It implies that a number (a) multiplied by 4 in mod 5 equals 3. From the table check for 4 on the residues column and 3 on the residues row

$$\scriptsize \therefore a = 2$$

i.e. 3 ⨸ 4 = 2 (mod 5)

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