The Madhura Fruits Company is packing four types of fruits into boxes. There are 126 oranges, 162 apples, 198 guavas and 306 pears. The fruits must be packed in such a way that a given box must have only one type of fruit and must contain the same number of fruit units as any other box.

What is the minimum number of boxes that must be used?

The number of oranges, apples, guavas, and pears = 126, 162, 198, and 306.

Each box must contain an equal number of fruits with only one type of fruit. The additional condition provided is that there should be a minimum number of boxes in total.

The distribution is possible in multiple ways in such a way that distribution in each box is placed in such that each box contains a certain number of fruits n which is a factor for all the four given number of fruits :

Arrangement of 1 fruit of one kind in a basket.

2 is a factor of 126, 162, 198, and 306. So we can place 2 fruits of a particular kind in a basket.

Since we were asked for the minimum number of boxes this is possible when a maximum number of fruits of a kind are placed in a box.

Hence each box must contain the Highest common factor for the four numbers :

The prime factorization for the four numbers :

126 : $$2\cdot7\cdot9$$, 162 : $$2\cdot9\cdot9$$, 198 : $$2\cdot9\cdot11$$, $$306\ =\ 2\cdot9\cdot17$$

The HCF is 18.

The number of boxes required for each :

$$\frac{126}{18},\ \frac{162}{18},\ \frac{198}{18},\ \frac{306}{18}$$

7+9+11+17 = 44.