Step 1: Understand what a multiplicative inverse is. It is a number that, when multiplied by the original number, gives a result of 1, under a certain modulus.
Step 2: Identify the number we are working with, which is 3, and the modulus, which is 7.
Step 3: We need to find a number 'x' such that (3 * x) mod 7 = 1.
Step 4: Test different values for 'x' starting from 1 up to 6 (since we are working modulo 7).
Step 5: Check x = 1: (3 * 1) mod 7 = 3 mod 7 = 3 (not 1).
Step 6: Check x = 2: (3 * 2) mod 7 = 6 mod 7 = 6 (not 1).
Step 7: Check x = 3: (3 * 3) mod 7 = 9 mod 7 = 2 (not 1).
Step 8: Check x = 4: (3 * 4) mod 7 = 12 mod 7 = 5 (not 1).
Step 9: Check x = 5: (3 * 5) mod 7 = 15 mod 7 = 1 (this is the answer!).
Step 10: Conclude that the multiplicative inverse of 3 modulo 7 is 5.
