Let n GE 2 and x 0 mod n The inverse of x mod n is a number

Let n GE 2 and x 0 mod n. The inverse of x mod n is a number y such that x * y = 1 mod n. An inverse may not always exist; when it exists, we denote it by x^-1 mod n. Compute the inverses of 1, ,6 mod 7. Prove that if n is prime, then any x = 0 mod n has an inverse. Show that then, as the domain and range have the same size, it then must also be surjective; in particular, 1 is in its image. Prove that if n is not prime, then there exists some x = 0 mod n which has no inverse.

Solution

I am solving the a-part since it contains 6 sub-parts for the calculation for n=1,2,3,4,5,6. Please post multiple problems to get the remaining answers

a)

Inverse of 1 mod 7

1 * x = 1 mod 7

x = 8 (since 8 mod 7 will be equal to 1)

Inverse of 2 mod 7

2 * x = 1 mod 7

x = 4 (since 8 mod 7 will be equal to 1)

Inverse of 3 mod 7

3 * x = 1 mod 7

x = 5 (since 15 mod 7 will be equal to 1)

Inverse of 4 mod 7

4 * x = 1 mod 7

x = 2 (since 8 mod 7 will be equal to 1)

Inverse of 5 mod 7

5 * x = 1 mod 7

x = 3 (since 15 mod 7 will be equal to 1)

Inverse of 6 mod 7

6 * x = 1 mod 7

x = 6 (since 36 mod 7 will be equal to 1)