10. (USA 6) ${ }^{\mathrm{IMO}}$ Find the largest number obtainable as the product of positive integers whose sum is 1976.
10. Let $a_{1}a_{i}$. Since $4=2^{2}$, we may assume that all $a_{i}$ are either 2 or 3, and $M=2^{k} 3^{l}$, where $2 k+3 l=1976$. (4) $k \geq 3$ does not yield the maximal value, since $2 \cdot 2 \cdot 2 < 3 \cdot 3$. Hence $k \leq 2$ and $2 k \equiv 1976 \pmod{3}$ gives us $k=1, l=658$ and $M=2 \cdot 3^{658}$.