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- Let and be positive integers. If and is the least common multiple of and , prove that . Note that it follows that the least common multiple of two positive relatively prime integers is their product.Let a be an integer. Prove that 3|a(a+1)(a+2). (Hint: Consider three cases.)Prove that if and are real numbers such that , then there exist a rational number such that . (Hint: Use Exercise 25 to obtain such that . Then choose to be the least integer such that . With these choices of and , show that and then that .) If and are positive real numbers, prove that there exist a positive integer such that . This property is called Archimedean Property of the real numbers. (Hint: If for all , then is an upper bound for the set . Use the completeness property of to arrive at a contradiction.)