In this exercise you will prove a version of the Chinese Remainder Theorem for three congruences. Let m1,m2,m3 be positive integers such that each pair is relatively prime. That is, gcd(m1,m2) = 1 and gcd(m1,m3) = 1 and gcd(m2,m3) = 1.
Let a1, a2, a3 be any three integers. Show that there is exactly one integer x in the interval 0 ≤ x
x ≡ a1 (mod m1), x≡ a2 (mod m2), x≡ a3 (mod m3).
Can you figure out how to generalize this problem to deal with lots of congruences
x ≡ a1 (mod m1), x≡ a2 (mod m2), . . . , x ≡ ar (mod mr)?
In particular, what conditions do the moduli m1,m2, . . . , mr need to satisfy?
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