34𝑥 ≡ 77 (𝑚𝑜𝑑 89)

To solve the equation 34𝑥 ≡ 77 (𝑚𝑜𝑑 89), we need to find the value of 𝑥 that satisfies the congruence.

Step 1: Find the modular inverse of 34 (mod 89). To find the modular inverse of 34 (mod 89), we need to find a number 𝑎 such that 34𝑎 ≡ 1 (mod 89). In other words, we need to find a number 𝑎 such that 34𝑎 - 1 is divisible by 89.

Using the Extended Euclidean Algorithm, we can find that the modular inverse of 34 (mod 89) is 26. This means that 34 * 26 ≡ 1 (mod 89).

Step 2: Multiply both sides of the congruence by the modular inverse. Multiplying both sides of the congruence 34𝑥 ≡ 77 (mod 89) by 26 (the modular inverse of 34), we get: 34𝑥 * 26 ≡ 77 * 26 (mod 89) This simplifies to: 884𝑥 ≡ 2002 (mod 89)

Step 3: Simplify the congruence. To simplify the congruence, we can reduce the coefficients to their smallest positive residues modulo 89. We can do this by taking the remainder when dividing by 89.

The congruence 884𝑥 ≡ 2002 (mod 89) becomes: 17𝑥 ≡ 35 (mod 89)

Step 4: Find the solution for 𝑥. To find the solution for 𝑥, we need to find a number 𝑏 such that 17𝑏 ≡ 35 (mod 89). In other words, we need to find a number 𝑏 such that 17𝑏 - 35 is divisible by 89.

Using trial and error or other methods, we find that 𝑏 = 2 is a solution. This means that 17 * 2 ≡ 35 (mod 89).

Therefore, the solution to the congruence 34𝑥 ≡ 77 (mod 89) is 𝑥 ≡ 2 (mod 89).