A. Let a, b, c be positive integers and suppose that

a | c, b | c, and gcd(a, b)=1.

Prove that ab | c.

b. Let x = c and x = c' be two solutions to the system of simultaneous congruences in the Chinese remainder theorem. Prove that

c= c' (mod m1m2mk)

Step-by-step explanation:

a|c means that c=a*k k is some positive integer. We know that b|c so b| ak and (a,b)=1, so it must be b|k, i.e k=b*r, r is some positive integer number. Now we have that c=abr, so ab| c.

B) if x and x’ are both solution then we have that

mi | x-x’ for every i.

By a) we have that m1m2...mk| x-x’, so x and x’ are equal by mod od m1m2...mk.

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