29. Let $a, b$ and $c$ be integers with $a>b>c>0$. If $b$ and $c$ are relatively prime, $b+c$ is a multiple of $a$, and $a+c$ is a multiple of $b$, determine the value of $a b c$.
29. Answer: 6
We shall show that $a=3, b=2$ and $c=1$.
Note that $2 a>b+c$. As $b+c$ is a multiple of $a$, it follows that $a=b+c$.
Let $a+c=k b$. Then $k b=a+c=b+c+c$, so $2 c=(k-1) b$. Since $c<b$, we must have $k=2$ and therefore $b=2 c$. Since $b$ and $c$ are relatively prime, this implies that $c=1$ and $b=2$. Thus $a=3$. Hence $a b c=6$.