System of linear congruences Help, please

[MathExpert]

5x == 200 (mod 251)
11x == 192 (mod 401)
3x == -151 (mod 907)

 

[matt grime]

first find the inverses of 5, 11 and 3 modulo 251, 401, 907 resp. to make it easier to solve. do this using eulcid's algorthim if you do'nt see what the inverse are straigh away. hint 5*50=250, and 3*302=906.

then use the chinese remainder theorem on the first two to reduce to two equivalences, then repeat.

 

[M98Ranger]

A system of linear congruences is a set of equations that involve modular arithmetic, where the unknown variable is congruent to a given value modulo a certain number. In this case, the system involves three equations with the unknown variable x, and each equation is congruent to a different value modulo a different number.

To solve this system, we can use the Chinese Remainder Theorem, which states that if the moduli (in this case, 251, 401, and 907) are pairwise coprime (meaning they have no common factors), then there exists a unique solution to the system.

First, we can simplify each equation by dividing both sides by the coefficient of x. This will give us the following system:

x == 40 (mod 251)
x == 192 (mod 401)
x == 252 (mod 907)

Next, we can use the Extended Euclidean Algorithm to find the inverse of each modulus. This will give us the following values:

251^-1 = 4 (mod 401)
401^-1 = 201 (mod 251)
907^-1 = 253 (mod 251)

Using these values, we can now apply the Chinese Remainder Theorem to solve the system. The solution will be given by the following formula:

x = (40*401*201 + 192*251*4 + 252*907*253) (mod 251*401*907)

Simplifying this, we get:

x = 5 (mod 907)

Therefore, the solution to this system of linear congruences is x = 5 (mod 907). I hope this helps!