Question

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?

• A. 21
• B. 42
• C. 36
• D. 18
• E. 44

Answer 1

Answer: (E)

To determine the minimum number of boxes needed to pack the fruits such that each box contains only one type of fruit and each box of a particular fruit type has the same number of fruits, we need to find the greatest common divisor (GCD) of the number of each type of fruit.

The numbers of fruits given are 126 oranges, 162 apples, 198 guavas, and 306 pears. To find the GCD, we perform the following steps:

1. List the prime factors of each number:
   • 126: The prime factors are \(2 \times 3^2 \times 7\).
   • 162: The prime factors are \(2 \times 3^4\).
   • 198: The prime factors are \(2 \times 3^2 \times 11\).
   • 306: The prime factors are \(2 \times 3^2 \times 17\).
2. Find the common factors among these numbers.
   • The common factors are \(2\) and \(3^2\) (as they appear in all sets of factors).
3. Thus, the GCD is calculated by multiplying these common factors:

The GCD of 126, 162, 198, and 306 is 18, which means that each box can contain 18 fruits of one type.

To find the minimum number of boxes needed, divide each type of fruit by the GCD:

• For oranges: \(126 \div 18 = 7\) boxes
• For apples: \(162 \div 18 = 9\) boxes
• For guavas: \(198 \div 18 = 11\) boxes
• For pears: \(306 \div 18 = 17\) boxes

Summing these results, the total number of boxes required is:

• \(7 + 9 + 11 + 17 = 44\) boxes

Therefore, the minimum number of boxes that must be used is 44.

Answer 2

To determine the minimum number of boxes needed, we must first find the greatest common divisor (GCD) of the number of each fruit type, as each box must contain an equal number of fruits of the same type. The fruits are:

• Oranges: 126
• Apples: 162
• Guavas: 198
• Pears: 306

The smallest number of fruits per box will be the GCD of these four numbers. Let's calculate the GCD step-by-step:

1. Find the GCD of 126 and 162:

• Prime factorization of 126: \(2 \times 3^2 \times 7\)
• Prime factorization of 162: \(2 \times 3^4\)
• Common factors: \(2 \times 3^2 = 18\)

2. Find the GCD of 18 and 198:

• Prime factorization of 198: \(2 \times 3^2 \times 11\)
• Common factors: \(2 \times 3^2 = 18\)

3. Find the GCD of 18 and 306:

• Prime factorization of 306: \(2 \times 3^2 \times 17\)
• Common factors: \(2 \times 3^2 = 18\)

The GCD of all four numbers is 18. Thus, each box will contain 18 fruits of one type.

Now, calculate the number of boxes for each fruit type:

Fruit Type	Total Fruits	Fruits per Box	Number of Boxes
Oranges	126	18	\(126 \div 18 = 7\)
Apples	162	18	\(162 \div 18 = 9\)
Guavas	198	18	\(198 \div 18 = 11\)
Pears	306	18	\(306 \div 18 = 17\)

Total number of boxes required = \(7 + 9 + 11 + 17 = 44\).

Thus, the minimum number of boxes that must be used is 44.