Step 2: Reduce 7 mod 5. Since 7 divided by 5 gives a remainder of 2, we can replace 7 with 2.
Step 3: Now the equation is 2x ≡ 3 (mod 5).
Step 4: To solve for x, we need to find the multiplicative inverse of 2 mod 5. This means we need a number that, when multiplied by 2, gives us 1 mod 5.
Step 5: Check the numbers 1, 2, 3, and 4 to find the inverse. 2 * 3 = 6, and 6 mod 5 = 1. So, the inverse of 2 mod 5 is 3.
Step 6: Multiply both sides of the equation 2x ≡ 3 (mod 5) by 3 (the inverse of 2). This gives us 3 * 2x ≡ 3 * 3 (mod 5).
Step 7: Simplify the left side: 3 * 2 = 6, and 6 mod 5 = 1. So, we have 1x ≡ 9 (mod 5).
Step 8: Simplify the right side: 9 mod 5 = 4. So, we have x ≡ 4 (mod 5).
Step 9: The solution x ≡ 4 (mod 5) means x can be 4, 9, 14, etc., but the smallest non-negative solution is 4.
Modular Arithmetic – Understanding how to solve equations in the context of modular arithmetic, including reducing coefficients and finding equivalent classes.
Linear Congruences – Solving linear congruences of the form ax ≡ b (mod m) and understanding the implications of the coefficients.
