When the four positive integers \(1\), \(1\), \(2\), and \(4\) are multiplied together, the result is \(8\) since \(1 \times 1 \times 2 \times 4=8\). When they are added together, the result is also \(8\) since \(1 + 1 + 2 + 4 = 8\).
We can also find a collection of \(n\) positive integers, with \(n>1\), that multiply to \(2020\) and add to \(2020\). Determine all possible values for \(n\).
To form such a collection of integers, our strategy is to determine all collections of integers larger than \(1\) whose product is \(2020\), and then for each collection add enough \(1\)s to make the sum of the numbers in the collection equal to \(2020\).
Since we want to consider integers whose product is \(2020\), we should find the divisors of \(2020\).
The prime factorization of \(2020\) is \(2 \times 2 \times 5 \times 101\). We can use this factorization to determine all collections of integers greater than \(1\) that multiply to \(2020\). The only collection with \(4\) integers is \(2,2,5,101\). The collections with \(3\) integers are \(4, 5, 101\), and \(2,10,101\), and \(2,5,202\), and \(2,2,505\). A systematic count determines that the only collections with \(2\) numbers are \(4,505\), and \(5,404\), and \(10,202\), and \(2,1010\), and \(20,101\). It is left for the reader to verify this. We will not consider the collection with just \(2020\), as it is given that the number of integers in the collection is to be greater than \(1\).
We list these collections in the first column of the table below. We also determine the sum of the integers in this collection, the number of \(1\)s needed (found by subtracting the sum from \(2020\)), and the value of \(n\), which is equal to the number of integers in first column plus the entry in third column.
| Collection of integers greater than \(1\) | Sum of integers greater than \(1\) in collection | Number of \(1\)s needed to sum to \(2020\) | Value of \(n\) |
|---|---|---|---|
| \(2,2,5,101\) | \(110\) | \(1910\) | \(1910 + 4 = 1914\) |
| \(4,5,101\) | \(110\) | \(1910\) | \(1910 + 3 = 1913\) |
| \(2,10,101\) | \(113\) | \(1907\) | \(1907 + 3 = 1910\) |
| \(2,5,202\) | \(209\) | \(1811\) | \(1811+ 3 = 1814\) |
| \(2,2,505\) | \(509\) | \(1511\) | \(1511 + 3 = 1514\) |
| \(4, 505\) | \(509\) | \(1511\) | \(1511 + 2 = 1513\) |
| \(5, 404\) | \(409\) | \(1611\) | \(1611 + 2 = 1613\) |
| \(10, 202\) | \(212\) | \(1808\) | \(1808 + 2 = 1810\) |
| \(2, 1010\) | \(1012\) | \(1008\) | \(1008 + 2 = 1010\) |
| \(20, 101\) | \(121\) | \(1899\) | \(1899 + 2 = 1901\) |
Therefore, the possible values for \(n\) are \(1010\), \(1513\), \(1514\), \(1613\), \(1810\), \(1814\), \(1901\), \(1910\), \(1913\), and \(1914\).