In modular arithmetic, what is the multiplicative inverse of 3 mod 11?

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Question: In modular arithmetic, what is the multiplicative inverse of 3 mod 11?

Options:

Correct Answer: 4

Solution:

The multiplicative inverse of 3 mod 11 is 4, since 3 * 4 ≡ 12 ≡ 1 (mod 11).

In modular arithmetic, what is the multiplicative inverse of 3 mod 11?

In modular arithmetic, what is the multiplicative inverse of 3 mod 11?

Step 1: Understand what a multiplicative inverse is. It is a number 'x' such that when you multiply it by another number (in this case, 3), the result is congruent to 1 under a certain modulus (here, 11).
Step 2: Set up the equation. We want to find a number 'x' such that 3 * x ≡ 1 (mod 11).
Step 3: Test different values for 'x' starting from 1 up to 10 (since we are working mod 11).
Step 4: Check x = 1: 3 * 1 = 3, which is not congruent to 1 mod 11.
Step 5: Check x = 2: 3 * 2 = 6, which is not congruent to 1 mod 11.
Step 6: Check x = 3: 3 * 3 = 9, which is not congruent to 1 mod 11.
Step 7: Check x = 4: 3 * 4 = 12. Now, find 12 mod 11, which is 1. So, 3 * 4 ≡ 1 (mod 11).
Step 8: Conclude that the multiplicative inverse of 3 mod 11 is 4.

Multiplicative Inverse – The multiplicative inverse of a number 'a' modulo 'm' is a number 'b' such that (a * b) mod m = 1.
Modular Arithmetic – A system of arithmetic for integers, where numbers wrap around after reaching a certain value (the modulus).