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# There are 96 apples and 112 oranges. These fruits are packed in boxes in such a way that each box contains fruits of the same variety, and every box contains an equal number of fruits. Find the minimum number of boxes in which all the fruits can be packed.

Last updated date: 14th Jul 2024
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Hint: We have to find the number of fruits in each box so that all the apples and oranges can be totally packed. Therefore, the minimum number of the required boxes should be a common factor of 96 and 112. However, as the minimum number of boxes required is to be found, we have to maximize the number of fruits in each box and therefore, we have to take the HCF of 96 and 112 as the number of fruits in each box. Thereafter we can count the number of boxes required for apples and oranges by dividing the total number of apples and oranges by the number of fruits in each box respectively.

We find that the prime factorization of 96 can be written as $96=2\times 2\times 2\times 2\times 2\times 3$
The prime factorization of 112 can be written as $112=2\times 2\times 2\times 2\times 7$
Thus the HCF of 112 and 96 is $2\times 2\times 2\times 2=16$.
Then, the number of boxes required for the apples is $\dfrac{96}{16}=6$
And the number of boxes required for the oranges is $\dfrac{112}{16}=7$
Thus the minimum number of total boxes required to fill the fruits should be $6+7=13$