There are 24 peaches, 36 apricots and 60 bananas and they have to be arranged in several rows in such a way that every row contains the same number of fruits of only one type. What is the minimum number of rows required for this to happen

Options

A. 12
B. 9
C. 10
D. 6

Answer

To solve the problem of arranging 24 peaches, 36 apricots, and 60 bananas in rows such that each row contains the same number of fruits of only one type, we need to find the highest common factor (HCF) of the three numbers. The HCF will give us the minimum number of rows required.

### Step-by-Step Solution:

1. **Identify the numbers of fruits**:
- Peaches: 24
- Apricots: 36
- Bananas: 60

2. **Find the prime factorization of each number**:
- For 24:
- \( 24 = 2^3 \times 3^1 \)
- For 36:
- \( 36 = 2^2 \times 3^2 \)
- For 60:
- \( 60 = 2^2 \times 3^1 \times 5^1 \)

3. **Identify the common prime factors**:
- The common prime factors among 24, 36, and 60 are 2 and 3.

4. **Find the lowest power of each common prime factor**:
- For the prime factor 2:
- The lowest power is \( 2^2 \) (from 36 and 60).
- For the prime factor 3:
- The lowest power is \( 3^1 \) (from 24 and 60).

5. **Calculate the HCF**:
- HCF = \( 2^2 \times 3^1 = 4 \times 3 = 12 \)

6. **Determine the minimum number of rows**:
- The minimum number of rows required is equal to the HCF, which is 12.

### Final Answer:
The minimum number of rows required is **12**.

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