\[\begin{array}{l}
{\text{In the product 1}} \times {\text{2}} \times 3 \times 4 \times 5 \times 6,{\text{ which one of the six numbers should}}\\
{\text{be increased by 1 to cause the greatest increase in the product?}}
\end{array}\]
\[\begin{array}{l}
{\text{Let }}x = {\text{1}} \times {\text{2}} \times 3 \times 4 \times 5 \times 6.\\
{\text{Then}}\\
{\text{2}} \times {\text{2}} \times 3 \times 4 \times 5 \times 6 = 2x;\\
{\text{1}} \times 3 \times 3 \times 4 \times 5 \times 6 = \frac{3}{2}x;\\
{\text{1}} \times 2 \times 4 \times 4 \times 5 \times 6 = \frac{4}{3}x\\
...
\end{array}\]
\[\begin{array}{l}
{\text{1}} \to {\text{2}}\\
{\text{greatest product is 2}} \times {\text{2}} \times 3 \times 4 \times 5 \times 6
\end{array}\]
