If 4x ≡ 1 (mod 9), what is the smallest positive integer solution for x?

If 4x ≡ 1 (mod 9), what is the smallest positive integer solution for x?

Step 1: Understand the equation 4x ≡ 1 (mod 9). This means we want to find a number x such that when we multiply it by 4 and then divide by 9, the remainder is 1.
Step 2: We need to find the multiplicative inverse of 4 modulo 9. This is a number that, when multiplied by 4, gives a result that is congruent to 1 modulo 9.
Step 3: Start testing small positive integers for x to see which one satisfies the equation. We will check x = 1, 2, 3, ..., until we find a solution.
Step 4: Check x = 1: 4 * 1 = 4, and 4 mod 9 = 4 (not a solution).
Step 5: Check x = 2: 4 * 2 = 8, and 8 mod 9 = 8 (not a solution).
Step 6: Check x = 3: 4 * 3 = 12, and 12 mod 9 = 3 (not a solution).
Step 7: Check x = 4: 4 * 4 = 16, and 16 mod 9 = 7 (not a solution).
Step 8: Check x = 5: 4 * 5 = 20, and 20 mod 9 = 2 (not a solution).
Step 9: Check x = 6: 4 * 6 = 24, and 24 mod 9 = 6 (not a solution).
Step 10: Check x = 7: 4 * 7 = 28, and 28 mod 9 = 1 (this is a solution!).
Step 11: Since we found that x = 7 satisfies the equation, we conclude that the smallest positive integer solution for x is 7.

Modular Arithmetic – Understanding congruences and how to find solutions to equations in modular systems.
Multiplicative Inverse – Finding the multiplicative inverse of a number in a given modulus, which is essential for solving modular equations.