Question

Write the multiplication table for the set of integers modulo 5.

Answer 1

Hint: In this particular question use the concept that in modulo 5 we have to consider the integer numbers from 0 to 4 i.e. 5 is not included and use the concept that if in the multiplication table if the number is equal to or greater than 5 then divide the number with 5 and write the remainder in the table, so use these concepts to reach the solution of the question.

Complete step-by-step answer:
We have to write the multiplication table for the set of integers modulo 5.
So the set of integers we have to take is (0, 1, 2, 3, 4)
Now the multiplication table for these set of integers are given below

	0	1	2	3	4
0	0	0	0	0	0
1	0	1	2	3	4
2	0	2	4	6	8
3	0	3	6	9	12
4	0	4	8	12	16

Now as we see that in the multiplication table there are several numbers which are greater than 5 so we have to divide them by 5 and in place of these numbers we have to write the remainders so we have,

	0	1	2	3	4
0	0	0	0	0	0
1	0	1	2	3	4
2	0	2	4	$\dfrac{6}{5} = 1\dfrac{1}{5}$	$\dfrac{8}{5} = 1\dfrac{3}{5}$
3	0	3	$\dfrac{6}{5} = 1\dfrac{1}{5}$	$\dfrac{9}{5} = 1\dfrac{4}{5}$	$\dfrac{{12}}{5} = 2\dfrac{2}{5}$
4	0	4	$\dfrac{8}{5} = 1\dfrac{3}{5}$	$\dfrac{{12}}{5} = 2\dfrac{2}{5}$	$\dfrac{{16}}{5} = 3\dfrac{1}{5}$

Where, $Q\dfrac{R}{D}$, Q = quotient, R = remainder, D = divisor.
Now write in place of the actual numbers only remainders so we have,

	0	1	2	3	4
0	0	0	0	0	0
1	0	1	2	3	4
2	0	2	4	1	3
3	0	3	1	4	2
4	0	4	3	2	1

So this is the required multiplication table for the set of integers modulo 5.
So this is the required answer.


Note: Whenever we face such types of questions the key concept we have to remember is that if there are 4 numbers say A, B, C and D then the first row of the multiplication table is formed by the multiplication of A with all of the numbers written in order, second row multiplication of B with all of the numbers written in order similarly for rest of the numbers.