Question

Find the inverse of 12 modulo 19

Answer 1

8 is the modular inverse of 12 mod 19 since 12*8 mod 19 ≡ 1.

Explanation:

We need to find the inverse of 12 modulo 19.

`12^(-1)(\text{ mod 19})`

If a is an integer and m is modulo, then the modular multiplicative inverse of a modulo m is an integer b such that

`a* b\equiv 1(\text{ mod m})`

Substitute different values of b and check whether that remainder is 1 after modulo 19.

At b=1,

`12* 1\equiv 12(\text{ mod 19})`

At b=2,

`12* 2\equiv 5(\text{ mod 19})`

At b=3,

`12* 3\equiv 17(\text{ mod 19})`

At b=4,

`12* 4\equiv 10(\text{ mod 19})`

At b=5,

`12* 5\equiv 3(\text{ mod 19})`

At b=6,

`12* 6\equiv 15(\text{ mod 19})`

At b=7,

`12* 7\equiv 8(\text{ mod 19})`

At b=8,

`12* 8\equiv 1(\text{ mod 19})`

Therefore, 8 is the modular inverse of 12 mod 19 since 12*8 mod 19 ≡ 1.