Answer:
the LCM of 105, 126, and 147 is 18420.
Since the fruit boxes need to be equally packed and no fruit is left, the largest possible number of fruit boxes needed is 18420/105 = 176 fruit boxes.
Explanation:
To find the largest number of fruit boxes needed, we need to find the least common multiple (LCM) of 105, 126, and 147.
What is the least common multiple (LCM)?
[The least common multiple is the smallest number that is a multiple of all the given numbers. One way to find the LCM of three numbers is to first find the LCM of two of the numbers, and then find the LCM of the result with the third number.]
To find the LCM of 105 and 126, we can list the multiples of each number and look for the smallest number that is a multiple of both:
Multiples of 105: 105, 210, 315, 420, 525, 630, 735, 840, 945, 1050, 1155, 1260, ...
Multiples of 126: 126, 252, 378, 504, 630, 756, 882, 1008, 1134, 1260, 1386, 1512, ...
The smallest multiple that both 105 and 126 share is 1260. Therefore, the LCM of 105 and 126 is 1260.
Now we can find the LCM of 1260 and 147 by finding the smallest multiple of 1260 that is also a multiple of 147:
Multiples of 1260: 1260, 2520, 3780, 5040, 6300, 7560, 8820, 10080, 11340, 12600, 13860, 15120, ...
Multiples of 147: 147, 294, 441, 588, 735, 882, 1029, 1176, 1323, 1470, 1617, 1764, ...
The smallest multiple that both 1260 and 147 share is 18420. Therefore, the LCM of 105, 126, and 147 is 18420.
Since the fruit boxes need to be equally packed and no fruit is left, the largest possible number of fruit boxes needed is 18420/105 = 176 fruit boxes.