Final answer:
The multiplicative inverse of 11 in Z17 is 8. This is found by using the congruence 45≡11 (mod 17) and calculating that the inverse of 45, which is -9, also serves as the inverse of 11, and the least positive residue modulo 17 is 8.
Step-by-step explanation:
To find the multiplicative inverse of 11 in Z17, we need to find a number which, when multiplied by 11, gives a product of 1 modulo 17. From the hint provided, we see that 45 is congruent to 11 modulo 17, that is, 45≡11 (mod 17). Knowing this, we can find a multiple of 45 that is one more than a multiple of 17, so that the inverse can be calculated easily.
Checking multiples of 45, we find that 45 x 3 = 135, which is equivalent to -2 modulo 17 (since 135-136=-1 and 136 is a multiple of 17). So, the inverse of -2 is -9 because -2 x -9 = 18 which is equivalent to 1 modulo 17. But we need the inverse of 11, not -2. Since 45≡11 and -2≡135 modulo 17, the inverse of 11 modulo 17 should be the same as the inverse of 45, which is -9. And because we typically represent numbers in their least positive residue form when working with modular arithmetic, we find the equivalent positive value for -9 modulo 17 by adding 17 to it, giving us 8.
Therefore, the multiplicative inverse of 11 in Z17 is 8.