Answer:
To solve this system of congruences, we can use the Chinese Remainder Theorem. We begin by finding the values of the constants that we will use in the CRT.
First, we have:
x ≡ 12 (mod 25)
This means that x differs from 12 by a multiple of 25, so we can write:
x = 25k + 12
Next, we have:
x ≡ 9 (mod 26)
This means that x differs from 9 by a multiple of 26, so we can write:
x = 26m + 9
Finally, we have:
x ≡ 23 (mod 27)
This means that x differs from 23 by a multiple of 27, so we can write:
x = 27n + 23
Now, we need to find the values of k, m, and n that satisfy all three congruences. We can do this by substituting the expressions for x into the second and third congruences:
25k + 12 ≡ 9 (mod 26)
This simplifies to:
k ≡ 23 (mod 26)
26m + 9 ≡ 23 (mod 27)
This simplifies to:
m ≡ 4 (mod 27)
We can use the first congruence to substitute for k in the second congruence:
25(23t + 12) ≡ 9 (mod 26)
This simplifies to:
23t ≡ 11 (mod 26)
We can solve this congruence using the extended Euclidean algorithm or trial and error. We find that t ≡ 3 (mod 26) satisfies this congruence.
Substituting for t in the expression for k, we get:
k = 23t + 12 = 23(3) + 12 = 81
Substituting for k and m in the expression for x, we get:
x = 25k + 12 = 25(81) + 12 = 2037
x = 26m + 9 = 26(4) + 9 = 113
x = 27n + 23 = 27(n) + 23 = 2037
We can check that all three of these expressions are congruent to 2037 (mod 25), 9 (mod 26), and 23 (mod 27), respectively. Therefore, the solution to the system of congruences is:
x ≡ 2037 (mod 25 x 26 x 27) = 14152