Paulinho was studying the Greatest Common Divisor (GCD) at school and decided to practice at home. He called the ages of three people living with him $a, b$, and $c$. Then, he performed some operations with their prime factors and obtained the greatest common divisors of the 3 pairs of numbers. A few days later, he forgot the ages $a, b$, and $c$, but found the following results noted:

$$
\begin{aligned}
a \cdot b \cdot c & =2^{4} \cdot 3^{2} \cdot 5^{3} \\
G C D(a, b) & =15 \\
G C D(a, c) & =5 \\
G C D(b, c) & =20
\end{aligned}
$$

Help Paulinho determine the values of $a, b$, and $c$.
Solution

Analyzing the greatest common divisors listed, we can ensure that $a$ is a multiple of 15 and of $5, b$ is a multiple of 15 and of 20, and $c$ is a multiple of 5 and of 20. Using the prime factors of these numbers, we have that $a$ is a multiple of $15, b$ is a multiple of 60, and $c$ is a multiple of 20. Thus, there exist positive integers $x, y$, and $z$, such that

$$
\begin{aligned}
a & =15 \cdot x=3 \cdot 5 \cdot x \\
b & =60 \cdot y=2^{2} \cdot 3 \cdot 5 \cdot y \\
c & =20 \cdot z=2^{2} \cdot 5 \cdot z
\end{aligned}
$$

Substituting these expressions for $a, b$, and $c$ into the product, we have $x \cdot y \cdot z=1$ and, therefore, $x=y=z=1$. Hence, $a=15, b=60$, and $c=20$.

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