Answer:
8 is the modular inverse of 12 mod 19 since 12*8 mod 19 ≡ 1.
Step-by-step explanation:
We need to find the inverse of 12 modulo 19.
[tex]12^{-1}(\text{ mod 19})[/tex]
If a is an integer and m is modulo, then the modular multiplicative inverse of a modulo m is an integer b such that
[tex]a\times b\equiv 1(\text{ mod m})[/tex]
Substitute different values of b and check whether that remainder is 1 after modulo 19.
At b=1,
[tex]12\times 1\equiv 12(\text{ mod 19})[/tex]
At b=2,
[tex]12\times 2\equiv 5(\text{ mod 19})[/tex]
At b=3,
[tex]12\times 3\equiv 17(\text{ mod 19})[/tex]
At b=4,
[tex]12\times 4\equiv 10(\text{ mod 19})[/tex]
At b=5,
[tex]12\times 5\equiv 3(\text{ mod 19})[/tex]
At b=6,
[tex]12\times 6\equiv 15(\text{ mod 19})[/tex]
At b=7,
[tex]12\times 7\equiv 8(\text{ mod 19})[/tex]
At b=8,
[tex]12\times 8\equiv 1(\text{ mod 19})[/tex]
Therefore, 8 is the modular inverse of 12 mod 19 since 12*8 mod 19 ≡ 1.
