8. In mod 11 arithmetic what is the additive inverse of 4? Answer: b. In mod 11 arithmetic what is the multiplicative inverse of 72 Answer: 7 - Show all steps in solving the congruency equation in mod 11. 7x + 4 = 11 5 Answer: x -

a) the additive inverse of 4 mod 11 is 7.

b) The multiplicative inverse of 72 mod 11 is 2.

c) x = 5

Step-by-step explanation:

The additive inverse of a number a in mod b is a number c such that a + c = 0 (mod b)

We need to find a number b such that 4 + b = 0 (mod 11)

4 + b = 0 (mod 11)

4 + b = 11k (mod 11) (if it's 0 (mod 11) then it's multiple of 11 and can be written as 11k

b = 11k - 4 (mod 11)

b = - 4 (mod 11)

b = 7 (mod 11)

Therefore, the additive inverse of 4 mod 11 is 7.

b) In mod 11 arithmetic what is the multiplicative inverse of 72?

The multiplicative inverse of a number a in mod b is a number c such that

ac = 1 mod b

First we see that 72 is larger than 11, 72/11 = 6 (11) + 6.

Therefore 72 = 6 (mod 11)

We need to find a number c such that 6c = 1 (mod 11)

This means we need to find a number c such that:

6c + 11a = 1 and this c will be the multiplicative inverse of 6.

We divide 11 by 6 and we get

11 = 6 (1) + 5

Now we divide 6 by 5 and we get:

6 = 5 (1) + 1

Therefore

1 = 6 - 5 (1)

But we knew that 5 is 11 - 6 (we are going to work back the other way)

1 = 6 - (11-6) (1)

1 = 6 - 11 (1) + 6 (1)

1 = - 11 (1) + 2 (6)

1 = 2 (6) - 11 (1) = 6 (2) + 11 (-1)

Thus c = 2 and this is the multiplicative inverse of 72.

c) Show all steps in solving the congruency equation in mod 11. 7x + 4 = 115

7x + 4 = 115 (mod 11)

We divide 115/11 and we get 115 = 11 (10) + 5.

So we can rewrite the equation to: 7x + 4 = 5 (mod 11)

7x + 4 = 5 (mod 11)

7x = 5 - 4 = 1 (mod 11)

7x = 1 (mod 11)

Therefore we need to find the multiplicative inverse of 7 mod 11.

7 x 1 = 7 (mod 11)

7 x 2 = 14 = 3 (mod 11)

7 x 3 = 21 = 10 (mod 11)

7 x 4 = 28 = 6 (mod 11)

7 x 5 = 35 = 2 (mod 11)

7 x 6 = 42 = 9 (mod 11)

7 x 7 = 49 = 5 (mod 11)

7 x 8 = 56 = 1 (mod 11)

7 x 9 = 63 = 8 (mod 11)

7 x 10 = 70 = 4 (mod 11)

Thus, the multiplicative inverse of 7 mod 11 is 8. and the solution of 7x + 4 = 115 (mod 11) is x = 8