## 38. How old is the eldest brother?

Determine this yourself, if it is known that the age of the middle brother is equal to the product of the ages of his two brothers, that the sum of the ages of all three brothers is 35, while the sum of the decimal logarithms of their ages is 3.
38. Let $a$ be the age of the oldest, $c$ the age of the middle, and $b$ the age of the youngest brother. Then, based on the conditions of the problem, the following relationships hold:

$$
\begin{gathered}
c^{2}=a b \\
a+b+c=35 \\
\lg a+\lg b+\lg c=3
\end{gathered}
$$

From the first equation, it follows that

$$
2 \lg c=\lg a+\lg b
$$

However, in this case, the third equation implies that

$$
\lg c=1 . \quad
$$

Thus, $c=10$, i.e., the middle brother is 10 years old. Therefore,

$$
a b=100 \text { and } a+b=25 .
$$

Hence, $a$ and $b$ are the roots of the equation

$$
x^{2}-25 x+100=0
$$

Since $a$ is greater than $b$, then $a=20, b=5$. Therefore, the age of the oldest brother is 20 years. (Since the age of the youngest brother $b=5$ years, indeed, $\lg a+\lg b+$ $+\lg c=\lg (a b c)=\lg (20 \cdot 5 \cdot 10)=\lg 1000=3$).