Final answer:
To find the modular inverse of a number, we need to find another number that multiplies with it to get a product that is equivalent to 1 modulo another number.
Step-by-step explanation:
The question is asking us to find the modular inverses of certain numbers. To find a modular inverse of a number a modulo m, we are looking for a number x such that (a * x) ≡ 1 (mod m). This is not the same concept as a multiplicative inverse in real numbers, which is simply ⅟/a.
For the first part, to find 1/11 mod 14, we need a number x such that (11 * x) ≡ 1 (mod 14). Testing numbers, we find that 13 works because (11 * 13) = 143, which leaves a remainder of 1 when divided by 14, hence 143 ≡ 1 (mod 14).
For the second part, to compute 1/7 mod 12, we search for a number y such that (7 * y) ≡ 1 (mod 12). Upon examination, 1 itself is the modular inverse since (7 * 1) = 7, which leaves a remainder of 7 when divided by 12, but there is no value of y that will result in a remainder of 1 in this case as 7 and 12 are not co-prime.
Lastly, for 1/2 mod 15, we want z such that (2 * z) ≡ 1 (mod 15). Number 8 is the modular inverse here because (2 * 8) = 16, and 16 leaves a remainder of 1 when divided by 15, so 16 ≡ 1 (mod 15).