Find all solutions of this system: x is congruent to 2 (mod 3), 2x is congruent to 3 (mod 5) and 3x is congruent to 4 (mod 7).

It is possible to solve this system using the
Linear Congruence Theorem and the Extended Euclidean
Algorithm.

We must solve the following system of
linear congruences:

3)

(mod 7)

Conveniently, the first congruence is already
written in terms of x. We can alternatively write this as x = 3k + 2.
We will substitute of x into our next
equation:

implies 6k -= -1 (mod 5)

Then solving for k using the
technique described on the Linear congruence theorem reference page, we find 
(mod 5)k = 5l + 4

Plugging this
back into our equation for x, we find

4) + 2 = 15l + 12 + 2= 15l + 14.

We can then plug this
into our last equation.

14) -= 4(mod 7) implies 45l -= -38 (mod 7)

Then solving
for l, we find  , or 
.

But we know x = 5l + 5, so plugging this back in we find
that

.

Therefore, the solution to the system is
-= 10 (mod 35)