1. Determine an inverse of a modulo m for a = 6 and m = 11. This...

1. Determine an inverse of a modulo m for a = 6 and m = 11. This is equivalent to answering the question “_______ is the unique inverse of 6 (mod 11) that is non-negative and < 11.” Show your work following the steps.

Beside the inverse you identified in part a), identify two other inverses of 6 (mod 11).

Hint: All of these inverses are congruent to each other mod 11.

Although the congruence can be solved using any of the inverses you identified in parts a) and b), use the unique inverse of 6 (mod 11) that you identified in part a), that is, the non-negative inverse of 6 (mod 11) that is < 11, to solve the following congruence. Show your work. 6x ≡ 8(mod 11).
Determine if the congruence 6x ≡ 11(mod 8) has a solution.

If there is a solution, identify a value for x. If there is no solution, explain why not.

Solutions

Expert Solution

(d)

Want to find the solution of the equation

Thus this congruence equation's solution doesnt not exist.

Another approach for not existence:

Since 6x - 8 k = 11,

Here 6x is an even number and 8k is also an even number , so is 6x - 8k, but right hand side, 11 is an odd number.

Since an odd number is not equal to any even number, hence this equation doesn't have solution.

Colby Messinger answered 2 years ago

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