Answer:
Explanation:
To solve the congruence 5x ≡ 6 (mod 7), we need to find a value for x that satisfies the equation.
First, we'll find the multiplicative inverse of 5 modulo 7. The multiplicative inverse of a modulo m is a number x such that ax ≡ 1 (mod m). In this case, we need to find a number x such that 5x ≡ 1 (mod 7).
To find the multiplicative inverse, we can try different values of x and compute the result until we find one that satisfies the equation. Alternatively, we can use a more efficient method like the Extended Euclidean Algorithm.
Using the Extended Euclidean Algorithm, we find that the multiplicative inverse of 5 modulo 7 is 3, as 5 * 3 ≡ 1 (mod 7).
Now, we can multiply both sides of the congruence by the multiplicative inverse (3) to isolate x:
5x ≡ 6 (mod 7)
3 * 5x ≡ 3 * 6 (mod 7)
15x ≡ 18 (mod 7)
Reducing the coefficients:
1 * x ≡ 4 (mod 7)
Since x ≡ 4 (mod 7), the solution to the congruence 5x ≡ 6 (mod 7) is x ≡ 4 (mod 7). This means that x can be written as x = 4 + 7k, where k is any integer.