19𝑥 ≡ 4 (𝑚𝑜𝑑 141)

To solve the equation 19𝑥 ≡ 4 (𝑚𝑜𝑑 141), we need to find the value of 𝑥 that satisfies the congruence.

Step 1: Find the modular inverse of 19 modulo 141. To find the modular inverse, we need to find a number 𝑎 such that (19𝑎) ≡ 1 (𝑚𝑜𝑑 141). In other words, we need to find a number 𝑎 such that (19𝑎) % 141 = 1.

Using the Extended Euclidean Algorithm, we can find that the modular inverse of 19 modulo 141 is 59. Therefore, we can rewrite the congruence as:

𝑥 ≡ (4 * 59) % 141

Step 2: Simplify the right-hand side of the congruence. Calculating (4 * 59) % 141, we get:

𝑥 ≡ 236 % 141

Step 3: Find the remainder of 236 divided by 141. Dividing 236 by 141, we get a quotient of 1 and a remainder of 95. Therefore, we can rewrite the congruence as:

𝑥 ≡ 95 (𝑚𝑜𝑑 141)

So, the solution to the congruence 19𝑥 ≡ 4 (𝑚𝑜𝑑 141) is 𝑥 ≡ 95 (𝑚𝑜𝑑 141).