Nia has four tomato plants, each with a different number of tomatoes on it. The product of the number of tomatoes on each plant is \(24\,300\). The total number of tomatoes on the three plants with the most tomatoes is \(43\), and the total number of tomatoes on the three plants with the fewest tomatoes is \(34\). Determine all possibilities for the number of tomatoes on each plant.
This problem is asking us to find four distinct positive integers with a product of \(24\,300\), such that the sum of the smallest three integers is \(34\) and the sum of the largest three integers is \(43\). We start by letting the four integers be \(a\), \(b\), \(c\), and \(d\). Since the sum of the smallest three integers is \(34\), it follows that \(a+b+c=34\). Since the sum of the largest three integers is \(43\), it follows that \(b+c+d=43\). Subtracting these two equations gives \(d-a=9\).
It would help if we had an upper bound for the smallest integer. Suppose \(a=12\). Then \(d=12+9=21\). The smallest product of four such distinct integers is \(12 \times 13 \times 14 \times 21 = 45\,864\). Since this is larger than \(24\,300\), it follows that \(a<12\).
Similarly, it would help to have a lower bound for the smallest integer. Suppose \(a=6\). Then \(d=6+9=15\). The largest product of four such distinct integers is \(6 \times 13 \times 14 \times 15 = 16\,380\). Since this is smaller than \(24\,300\), it follows that \(a>6\).
This leaves us with the following possibilities for \((a,d)\): \((7,16)\), \((8,17)\), \((9,18)\), \((10,19)\), or \((11,20)\). Next we can factor \(24\,300\) to obtain \(24\,300=2^2 \times 3^5 \times 5^2\). From the factorization we can eliminate some of these pairs. Since \(7\), \(17\), \(19\), and \(11\) are not factors of \(24\,300\), they cannot be any of the four integers. Thus, the only possibility left is \((a,d)=(9,18)\).
Since \(24\,300=2^2 \times 3^5 \times 5^2\), it follows that the middle two integers must have a product of \(2 \times 3 \times 5^2\). The only possibilities between \(9\) and \(18\) are \(10\) and \(15\).
Therefore, there is only one possibility. The number of tomatoes on each plant is \(9\), \(10\), \(15\), and \(18\).