Problem 3. If the number 860 is divided by some number, a remainder of 9 is obtained, and if the number 1200 is divided by the same number, a remainder of 16 is obtained. Determine the appropriate quotients?
Solution. Let $x$ be the number by which 860 and 1200 are divided, and let $k_{1}$ be the quotient obtained when 860 is divided by $x$, and $k_{2}$ be the quotient when 1200 is divided by $x$. We have,

$$
\begin{aligned}
& 860=x k_{1}+9 \\
& x k_{1}=851
\end{aligned}
$$

And

$$
\begin{aligned}
& 1200=x k_{2}+13 \\
& x k_{2}=1184
\end{aligned}
$$

Accordingly, $x$ is a common divisor of 851 and 1184, which is greater than 16. However,

$$
\begin{aligned}
\mathrm{GCD}(851,1184) & =\mathrm{GCD}(1184-851,851)=\mathrm{GCD}(333,851) \\
& =\operatorname{GCD}(333,851-333)=\operatorname{GCD}(333,518) \\
& =\operatorname{GCD}(333,518-333)=\operatorname{GCD}(333,185) \\
& =\mathrm{GCD}(333-185,185)=\mathrm{GCD}(148,185) \\
& =\mathrm{GCD}(148,185-148)=\mathrm{GCD}(148,37)=37
\end{aligned}
$$

since $37 \mid 148=37 \cdot 4$. Accordingly,

$$
860=37 \cdot 23+9 \text { and } 1200=37 \cdot 32+16 \text {, }
$$

which means that the quotient when 860 is divided by 37 is 23, and when 1200 is divided by 37 it is 32.