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 m1m2 ... mk)

Question

Georgie

Mathematics

30 April, 00:34

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 m1m2 ... mk)

+1

Answers (1)

Know the Answer?

Essence Williamson · Answer 1

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.

A. Let a, b, c be positive integers and suppose thata | 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 thatc = c' (mod m1m2 ... mk)

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 m1m2 ... mk)