Using the congruence solve the LDE 6𝑥 +9𝑦 = 15

To solve the linear Diophantine equation (LDE) 6𝑥 + 9𝑦 = 15 using congruence, we can follow these steps:

Step 1: Find the greatest common divisor (GCD) of the coefficients 6 and 9. In this case, the GCD of 6 and 9 is 3.

Step 2: Check if the GCD divides the constant term 15. Since 3 divides 15 evenly, we can proceed to the next step.

Step 3: Divide the equation by the GCD. Dividing both sides of the equation by 3, we get 2𝑥 + 3𝑦 = 5.

Step 4: Rewrite the equation in congruence form. We can rewrite the equation as 2𝑥 ≡ 5 (mod 3).

Step 5: Solve the congruence equation. To find the solution, we need to find the modular inverse of 2 modulo 3. The modular inverse of 2 modulo 3 is 2 itself, since 2 * 2 ≡ 1 (mod 3).

Step 6: Multiply both sides of the congruence equation by the modular inverse. Multiplying both sides by 2, we get 𝑥 ≡ 10 (mod 3).

Step 7: Simplify the congruence equation. Since 10 ≡ 1 (mod 3), we can simplify the congruence equation to 𝑥 ≡ 1 (mod 3).

Step 8: Substitute the value of 𝑥 into the original equation to find 𝑦. Substituting 𝑥 = 1 into the original equation 6𝑥 + 9𝑦 = 15, we get 6(1) + 9𝑦 = 15.

Step 9: Solve for 𝑦. Simplifying the equation, we have 6 + 9𝑦 = 15. Subtracting 6 from both sides, we get 9𝑦 = 9. Dividing both sides by 9, we find 𝑦 = 1.

Therefore, the solution to the LDE 6𝑥 + 9𝑦 = 15 is 𝑥 ≡ 1 (mod 3) and 𝑦 = 1.