Let's consider the number of factors each number from 1 to 10 has.
1 has 1 factor.
2 has 2 factors (1 and 2).
3 has 2 factors (1 and 3).
4 has 3 factors (1, 2, and 4).
5 has 2 factors (1 and 5).
6 has 4 factors (1, 2, 3, and 6).
7 has 2 factors (1 and 7).
8 has 4 factors (1, 2, 4, and 8).
9 has 3 factors (1, 3, and 9).
10 has 4 factors (1, 2, 5, and 10).
The number of factors each number has is related to the number of ways the number can be split into two groups such that the sum of the numbers in one group is equal to the number and the product of the numbers in the other group is equal to the number.
For example, consider the number 6. It can be split into the group [1, 5] and [2, 3], which add up to 6 and have a product of 6, respectively.
The largest number of factors that a number from 1 to 10 can have is 4, so the largest possible value of n is the number with the most factors, which is 6. Thus, the largest possible value of n is 6