The solution to the system of congruences
x = 1 (mod 12)
x = 10 (mod 21)
can be written as x = k (mod n), where k is a modulo-n residue.
a) find the value of k
b) find the value of n
*pretend that the equal sign in the congruence symbol
(a) Find the value of k
The Chinese Remainder Theorem states that for a system of congruences of the form
x ≡ a1 (mod m1) x ≡ a2 (mod m2) ... x ≡ an (mod mn)
where the moduli are pairwise relatively prime, there is a unique solution modulo the product of all the moduli.
In this case, the moduli 12 and 21 are pairwise relatively prime, so there is a unique solution modulo 12 * 21 = 252.
To find the solution, we can use the following steps:
Find the modular inverses of 12 and 21 modulo each other.
Compute the weighted sum of the congruences, using the modular inverses as weights.
Reduce the weighted sum modulo 252.
The modular inverse of 12 modulo 21 is 7, and the modular inverse of 21 modulo 12 is 4.
Therefore, the weighted sum of the congruences is:
7 * 1 (mod 21) + 4 * 10 (mod 12)
This simplifies to:
7 + 40
Reducing modulo 252, we get k = 47.
(b) Find the value of n
The value of n is the product of the moduli in the system of congruences, which is 252.
Therefore, the solution to the system of congruences can be written as x = 47 (mod 252).