4. Let S be a table consisting of positive integers - the table can contain the same numbers - and it contains the number 68. The mean (arithmetic average) of the numbers in S is 56. However, if 68 is removed, the mean of the remaining numbers drops to 55. What is the largest number that could appear in S?
Let $\mathrm{n}$ denote the number of positive integers in $\mathrm{S}$, then the sum of these $\mathrm{n}$ numbers is $56 n$. After removing 68, the sum of the remaining $n-1$ numbers is $55(n-1)$, thus we get $56 n-55(n-1)$ $=68$, solving for $\mathrm{n}$ gives $\mathrm{n}=13$. After removing 68, the sum of the remaining 12 numbers in $\mathrm{S}$ is $55 \times 12=660$. To make one of these 12 numbers the largest, the other numbers should be as small as possible. For this, $\mathrm{S}$ should have 11 ones, and the largest possible number would be $660-11=649$.