Construct a take for multiplication in Modulo 7 on the set {2,3,5,6}
Use the table to solve the following equation
1 a×a=2
2 b×(b×6)=3
To perform multiplication in Modulo 7 on the set {2,3,5,6}, we multiply two numbers as we normally would and then take the remainder when divided by 7.
For example, let's calculate 3 × 6 in Modulo 7:
3 × 6 = 18
18 divided by 7 equals 2 remainder 4
So, 3 × 6 in Modulo 7 is equal to 4.
Now, let's use this concept to solve the given equations:
1. a × a = 2
We need to find a number 'a' such that, when multiplied by itself in Modulo 7, the result is 2.
Let's try each number from the given set and calculate their squares in Modulo 7:
2 × 2 = 4 (Mod 7)
3 × 3 = 2 (Mod 7)
5 × 5 = 4 (Mod 7)
6 × 6 = 1 (Mod 7)
Hence, the solution to the equation a × a = 2 in Modulo 7 is a = 3.
2. b × (b × 6) = 3
We need to find a number 'b' such that, when multiplied by itself and then by 6 in Modulo 7, the result is 3.
Let's try each number from the given set and calculate their values in Modulo 7:
For b = 2:
2 × (2 × 6) = 24 (Mod 7)
24 divided by 7 equals 3 remainder 3
So, b = 2 is not a solution.
For b = 3:
3 × (3 × 6) = 54 (Mod 7)
54 divided by 7 equals 7 remainder 5
So, b = 3 is not a solution.
For b = 5:
5 × (5 × 6) = 150 (Mod 7)
150 divided by 7 equals 21 remainder 3
So, b = 5 is a solution.
For b = 6:
6 × (6 × 6) = 216 (Mod 7)
216 divided by 7 equals 30 remainder 6
So, b = 6 is not a solution.
Hence, the solution to the equation b × (b × 6) = 3 in Modulo 7 is b = 5.