A multiplications modulo 6 and 7 construct the following...

A. Multiplications Modulo 6, and 7 Construct the following multiplication tables; Mod 6 X 0 1 2 3 4 5 0 1 2 3 4 5 Mod 7 X 0 1 2 3 4 5 6 0 1 2 3 4 5 6 2. Observe any differences between these two tables. For example; in mod 6 : 2 x3 = 0 (mod 6) but in mod 7: 2x3 = 6 (mod 7) . Explain and express your own subtle observations here: Can you explain the number of rows and columns? Any subtle distinctions between the two tables? Any interesting facts in each table? Are there any special elements in each table? Are there any special pairs of elements? Can you observe any patterns or any behavior in these tables?

Transcript

00:01 In this problem we required to construct the following multiplication tables for modulo 6 and 7 and we have to write the observations based on the tables.

00:10 So let us start with the modulo 6.

00:12 Here 0 times 0 will be equal to 0, divided by 6 will give the remainder as 0.

00:18 So this is 0.

00:19 Then again 0 times 1 is 0 and all these will give a remainder 0 when divided by 6.

00:28 Next is 1 times 0 is 0 which will give remainder 0.

00:32 Then 1 times 1 divided by 6 give the remainder 1.

00:36 1 times 2 gives remainder 2.

00:38 1 times 3 gives remainder 3 when divided by 6.

00:41 And similarly this is 4 and 1 times 5 is 5 gives remainder 5 when divided by 6.

00:52 Again 2 times 0 is 0 divided by 6 gives remainder 0 and all these will become 0.

00:59 2 times 1 is 2 divided by 6 gives 2.

01:03 2 times 2 is 4 divided by 6.

01:05 It gives the remainder 0.

01:07 2 times 3 is 6 divided by 6 it gives the remainder 0 2 times 4 is 8 divided by 6 it gives the remainder 2 2 times 5 is 10 divided by 6 it gives the remainder 4 then we move to 3 3 times 1 3 times 1 2 is 6 divided by 6 it gives the remainder 0 3 times 3 times 3 is 9 it gives the remainder 3 when divided by 6 3 times 4 when divided by 6 it gives the remainder 0 3 times 5 is 15 divided by 6 it gives the remainder 3 4 times 1 is 4 divided by 6 it gives the remainder 4 times 2 is 8 divided by 6 it gives 2 as remainder 4 times 3 is 12 divided by 6 it gives the remainder 4 times 5 is 20 divided by 6 it gives the remainder 2 moving to 5, 5 times 1 is 5 divided by 6, it gives 5 is remainder, then 5 times 2 is 10 divided by 6 it gives 4 as remainder, 5 times 3 is 20 divided by 6 it gives 2 as remainder, and 5 times 2 is 20 divided by 6 it gives 1 as remainder.

02:28 Then we complete the table 4 module 7.

02:31 Here we have 0 times 0, it will give 0 as remainder when divided by 7.

02:35 And all this will come 0 and similarly the columns will also become 0 for 0.

02:43 Then completing the table in a similar manner we have 1, 2, 3, 4, 5, 6.

02:52 For the next row we have 2, 4, 6, 1, 3, 5, and 4.

03:08 6, 1 and 4.

03:10 For the next row we have 4 1 5 2 6 3...