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Solve the congruence 5x ≡ 6 (mod7).

______, m is any integer

User Gjeltema
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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
User Rob Van Der Veer
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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.

User Pizzafilms
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