Question

Find all the solutions to the following system of linear

congruences: x ≡ 1( mod 2), x ≡ 2( mod 3), x ≡ 3( mod 5), x ≡ 4(
mod 7), and x ≡ 5

Answer 1

To find the solutions to the given system of linear congruences, we solve each congruence individually and then find the intersection of all the solutions.

Step-by-step explanation:

To find the solutions to the given system of linear congruences, we need to solve each congruence individually and then find the intersection of all the solutions. Let's start with the first congruence x ≡ 1 (mod 2). This congruence implies that x is congruent to 1 mod 2, which means x is an odd number. The possible solutions for this congruence are 1, 3, 5, and so on.

Next, let's consider the second congruence x ≡ 2 (mod 3). This congruence means x is congruent to 2 mod 3, which implies that x leaves a remainder of 2 when divided by 3. The possible solutions for this congruence are 2, 5, 8, and so on.

Similarly, we can solve the remaining congruences x ≡ 3 (mod 5), x ≡ 4 (mod 7), and x ≡ 5. For x ≡ 3 (mod 5), the possible solutions are 3, 8, 13, and so on. For x ≡ 4 (mod 7), the possible solutions are 4, 11, 18, and so on. Finally, for x ≡ 5, the only possible solution is 5.

To find the intersection of all the solutions, we need to look for numbers that satisfy all the congruences. In this case, the common solutions to all the congruences are 5, 13, 21, and so on.