Final answer:
The multiplicative inverse of 5 in Z11 is 9, does not exist in Z12 as 5 and 12 are not coprime, and is 8 in Z13.
Step-by-step explanation:
The multiplicative inverse of a number n in a given modulo m system is a number which, when multiplied by n, yields 1 modulo m.
Essentially, it's the equivalent of division in modular arithmetic, often represented as n-1.
To find the multiplicative inverses of the number 5 in Z11, Z12, and Z13, we need to find numbers x, such that (5 * x) % 11 = 1, (5 * x) % 12 = 1, and (5 * x) % 13 = 1, respectively.
In Z11 (mod 11), the multiplicative inverse of 5 is 9 since (5 * 9) % 11 = 45 % 11 = 1.
In Z12 (mod 12), there is no multiplicative inverse for 5 because 5 and 12 are not coprime (they share a common factor of 1 other than 1).
In Z13 (mod 13), the multiplicative inverse of 5 is 8 since (5 * 8) % 13 = 40 % 13 = 1.