Answer:
below
Explanation:
To find the value of x in a congruence equation, you need to solve for x within the given modulus. Here are the general steps to follow:
Start with a congruence equation in the form of ax ≡ b (mod m), where a, b, and m are known integers.
Check if the equation has a solution. For a congruence equation ax ≡ b (mod m) to have a solution, the greatest common divisor (GCD) of a and m must divide b. If the GCD(a, m) does not divide b, the equation has no solution.
If the equation has a solution, proceed to find it. First, find the modular inverse of a modulo m. The modular inverse is the number x such that ax ≡ 1 (mod m). If the modular inverse exists, multiply both sides of the congruence equation by it.
Example: If you have 3x ≡ 2 (mod 7), the modular inverse of 3 modulo 7 is 5 because 3 * 5 ≡ 1 (mod 7). Multiply both sides of the congruence equation by 5: 5 * 3x ≡ 5 * 2 (mod 7), which simplifies to 15x ≡ 10 (mod 7).
Simplify the equation further by reducing the coefficients and values modulo m. In the previous example, 15x ≡ 10 (mod 7) can be simplified to 1x ≡ 3 (mod 7) since 15 ≡ 1 (mod 7) and 10 ≡ 3 (mod 7).
Solve for x. To find the value of x, divide both sides of the congruence equation by the coefficient of x (which is 1 in this case). In the example, x ≡ 3 (mod 7), meaning that x is congruent to 3 modulo 7. The value of x can be any integer that satisfies this congruence relation, such as x = 3, 10, 17, etc.
Remember that in congruence, there can be multiple solutions within the given modulus.