Jamal grew four tomato plants last summer. The product of the total number of tomatoes each plant produced is \(40\,392\). The plant that produced the most tomatoes produced exactly \(50\) more tomatoes than the plant that produced the fewest tomatoes. If the plant that produced the fewest tomatoes produced fewer than \(10\) tomatoes, determine all possibilities for the number of tomatoes each plant produced.
This problem is asking us to find four positive integers with a product of \(40\,392\), where the difference between the smallest and largest integer is \(50\), and the smallest integer is less than \(10\). We can start by factoring \(40\,392\). This gives \(40\,392=1 \times 2^3 \times 3^3 \times 11 \times 17\). We have included \(1\) since it could be one of the positive integers.
Since the smallest integer is less than \(10\), from the factorization we can see that it could be \(1\), \(2\), \(3\), \(4\), \(6\), \(8\), or \(9\). We will consider these cases.
If the smallest integer is \(1\), then the largest integer must be \(51\). Since \(51=3 \times 17\), it follows that \(51\) is a factor of \(40\,392\). The product of the other two integers must then be \(2^3 \times 3^2 \times 11 = 792\).
Thus, we are looking for two numbers, both less than \(51\), that multiply to \(792\). The factor pairs of \(792\) with both factors less than \(51\) are: \(18\) and \(44\), \(22\) and \(36\), and \(24\) and \(33\).
Therefore, the four integers could be
\(1\), \(18\), \(44\), and \(51\)
\(1\), \(22\), \(36\), and \(51\)
\(1\), \(24\), \(33\), and \(51\)
If the smallest integer is \(2\), then the largest integer must be \(52\). However \(52\) is not a factor of \(40\,392\), so this is not a possibility.
If the smallest integer is \(3\), then the largest integer must be \(53\). However \(53\) is not a factor of \(40\,392\), so this is not a possibility.
If the smallest integer is \(4\), then the largest integer must be \(54\). Since \(54=2 \times 3^3\), it follows that \(54\) is a factor of \(40\,392\). The product of the other two integers must then be \(11 \times 17\). Thus, the four integers could be \(4\), \(11\), \(17\), and \(54.\)
If the smallest integer is \(6\), then the largest integer must be \(56\). However \(56\) is not a factor of \(40\,392\), so this is not a possibility.
If the smallest integer is \(8\), then the largest integer must be \(58\). However \(58\) is not a factor of \(40\,392\), so this is not a possibility.
If the smallest integer is \(9\), then the largest integer must be \(59\). However \(59\) is not a factor of \(40\,392\), so this is not a possibility.
Therefore, there are four possibilities for the number of tomatoes each plant produced. They are as follows:
\(1\), \(18\), \(44\), and \(51\)
\(1\), \(22\), \(36\), and \(51\)
\(1\), \(24\), \(33\), and \(51\)
\(4\), \(11\), \(17\), and \(54\)