Question

Arrange the steps in the correct order to solve the system of congruences x=2 (mod 3), x=1 (mod 4), and x = 3 (mod 5) using the method of back substitution.

A.The first congruence can be written as x=3t+ 2 where t is an Integer. Substituting this expression for x into the second congruence gives 3t+ 2 = 1 (mod 4).
B.This implies t = 1 (mod 4). Therefore, t=4u+ 1 for some Integer u.
C,Thus, x=3t+2=3(4u+1)+2=12u+ 5. Substituting this in the third congruence to obtain 12u+ 5 = 3 (mod 5), which Implies u = 4 (mod 5).
D.Hence, u=5v+ 4 and so x=12u+5=12(5v+4)+5=60v+53, where vis an integer.
E.Translating x=60v+53 back into a congruence, we find the solution to the simultaneous congruences x = 53 mod 60.

Answer 1

To solve the system of congruences x=2 (mod 3), x=1 (mod 4), and x = 3 (mod 5) using the method of back substitution, follow these steps: 1. Rewrite the first congruence, 2. Substitute the expression for x into the second congruence, 3. Solve the congruence, 4. Replace t in the expression for x, 5. Substitute x into the third congruence and solve, 6. Replace u in the expression for x, 7. Translate x back into a congruence to find the solution.

Step-by-step explanation:

1. Rewrite the first congruence x=2 (mod 3) as x=3t+2.
2. Substitute the expression for x from step 1 into the second congruence, giving 3t+2 = 1 (mod 4).
3. Solve the congruence from step 2 to find t = 1 (mod 4), which implies t=4u+1 for some integer u.
4. Replace t in the expression from step 1 with 4u+1 to obtain x = 12u+5.
5. Substitute x=12u+5 into the third congruence x=3 (mod 5) and solve to find u = 4 (mod 5).
6. Replace u in the expression from step 4 with 5v+4 to obtain the final solution x = 60v+53.
7. Translate x=60v+53 back into a congruence to find the solution x = 53 (mod 60).