# Solution problem 2 about combinatoric in Shortlist problem in IMO 2016

**Problem Combinatoric 2**

Find all positive integers n for which all positive divisors of n can be put into the cells of a rectangular table under the following constraints:

- each cell contains a distinct divisor;
- the sums of all rows are equal; and
- the sums of all columns are equal.

Answer : 1.

Also Read:

**Solution 1. **

Suppose all positive divisors of n can be arranged into a rectangular table of size k x l where the number of rows k does not exceed the number of columns l. Let the sum of numbers in each column be *s*. Since n belongs to one of the columns, we have s > n, where equality holds only when n = 1.

For *j = 1,2,…l*, let d be the largest number in the *j-th* column. Without loss of generality, assume . Since these are divisors of n, we have

…………(1)

As d_{l} is the maximum entry of the l-th column, we must have

……………(2)

The relations (1) and (2) combine to give , that is, . Together with we conclude that k = l. Then all inequalities in (1) and (2) are equalities. In particular, s = n and so n = 1, in which case the conditions are clearly satisfied.

**Also Read : Shortlist problem Combinatorics in IMO 2016 ( international mathematic Olympiad)**