## 8. Common Measure and Multiple

Let $a, b$ and $c$ be natural numbers such that

$$
D(a, b)=4, \quad D(b, c)=6 \text { and } V(a, b, c)=36000
$$

What is the maximum value of $D(a, c)$?

$D(m, n)$ denotes the greatest common divisor of $m$ and $n$, and $V(p, q, r)$ denotes the least common multiple of $p, q$ and $r$.

Result: $\quad 250$
## Solution:

By factoring into prime factors, we get $36000=2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5 \cdot 5 \cdot 5$.

From the conditions of the problem, it follows that

$$
\begin{gathered}
a=2 \cdot 2 \cdot x \\
b=2 \cdot 2 \cdot 3 \cdot y \\
c=2 \cdot 3 \cdot z
\end{gathered}
$$

Where $x$ cannot be divisible by 3, and $z$ cannot be divisible by 2. It follows that $D(x, z)$ divides $5 \cdot 5 \cdot 5$ so $D(a, c)$ can be at most $2 \cdot 5 \cdot 5 \cdot 5=250$.