(1) The least common multiple of nonzero integers $\displaystyle a$ and $\displaystyle b$ is the smallest positive integer $\displaystyle m$ such that $\displaystyle a|m$ and $\displaystyle b|m$ ; $\displaystyle m$ is usually denoted $\displaystyle [a,b]$. Prove that:

(a) whenever $\displaystyle a|k$ and $\displaystyle b|k$, then $\displaystyle [a,b]|k$

(b) $\displaystyle [a,b] = ab/(a,b)$ if $\displaystyle a>0$ and $\displaystyle b>0$

(2) Prove that a positive integer is divisible by 3 if and only if the sum of its digits is divisible by 3. [Hint: $\displaystyle 10^3 = 999+1$ and similarly for other powers of 10.]

(3) Prove that for every $\displaystyle n, (n+1, n^2-n+1) $is $\displaystyle 1 or 3$

Thanks for any help!