(1) The least common multiple of nonzero integers and is the smallest positive integer such that and ; is usually denoted . Prove that:

(a) whenever and , then

(b) if and

(2) Prove that a positive integer is divisible by 3 if and only if the sum of its digits is divisible by 3. [Hint: and similarly for other powers of 10.]

(3) Prove that for every is

Thanks for any help!