Solve
X=8(mod 11), X=9(mod 12), X=10(mod 13)
52 = 8(mod 11)
57 = 9(mod 12)
62 = 10(mod 13)
To solve the system of congruences:
X ≡ 8 (mod 11)
X ≡ 9 (mod 12)
X ≡ 10 (mod 13)
We can use the Chinese Remainder Theorem (CRT) to find the unique solution modulo the product of the moduli. Here are the steps to solve the system:
Step 1: Find the product of the moduli.
M = 11 * 12 * 13 = 1716
Step 2: For each congruence, find the value of Mi, where Mi = M/mi and mi is the corresponding modulus. In this case:
M1 = 12 * 13 = 156
M2 = 11 * 13 = 143
M3 = 11 * 12 = 132
Step 3: Find the modular inverse of Mi modulo mi. The modular inverse of a number a modulo b is denoted as a^(-1) (mod b) and satisfies the congruence a * a^(-1) ≡ 1 (mod b).
Using modular inverse calculations or the extended Euclidean algorithm, we find:
M1^(-1) ≡ 9 (mod 11)
M2^(-1) ≡ 10 (mod 12)
M3^(-1) ≡ 5 (mod 13)
Step 4: Compute the sum of (Ai * Mi * Mi^(-1)) for each congruence, where Ai is the residue (right-hand side) of the corresponding congruence.
For the first congruence:
A1 = 8
A2 = 9
A3 = 10
Calculations:
X ≡ (8 * 156 * 9 + 9 * 143 * 10 + 10 * 132 * 5) (mod 1716)
Step 5: Simplify the value obtained in Step 4 modulo M to obtain the unique solution.
Calculating the expression above, we have:
X ≡ 10644 (mod 1716)
To find the smallest positive solution, we can subtract multiples of 1716 until we get a result between 0 and 1715.
X ≡ 10644 - (6 * 1716) (mod 1716)
X ≡ 14 (mod 1716)
Therefore, the solution to the system of congruences is:
X ≡ 14 (mod 1716)