There are `527` apples, `646` pears and `748` oranges . These are to be arranged in heaps containing the same number of fruits . Find the greatest number of fruits possible in each heap. How many heaps are formed ?

Options

A. Number of heaps: `16`
Greatest number of fruits possible: `163`
B. Number of heaps: `17`
Greatest number of fruits possible: `133`
C. Number of heaps: `19`
Greatest number of fruits possible: `123`
D. Number of heaps: `15`
Greatest number of fruits possible: `103`

Answer

To solve the problem of arranging `527` apples, `646` pears, and `748` oranges into heaps containing the same number of fruits, we need to find the greatest number of fruits possible in each heap. This can be done by calculating the Highest Common Factor (HCF) of the three numbers.

### Step-by-Step Solution:

1. **Identify the Numbers**: We have three quantities of fruits:
- Apples: 527
- Pears: 646
- Oranges: 748

2. **Find the Prime Factorization**:
- **Prime Factorization of 527**:
- 527 = 17 × 31
- **Prime Factorization of 646**:
- 646 = 2 × 17 × 19
- **Prime Factorization of 748**:
- 748 = 2 × 2 × 11 × 17

3. **Identify Common Factors**: From the prime factorizations:
- The common prime factor among all three numbers is `17`.

4. **Calculate the HCF**: The highest common factor (HCF) of `527`, `646`, and `748` is `17`.

5. **Determine the Number of Heaps**:
- **For Apples**:
- Number of heaps = Total apples / HCF = 527 / 17 = 31 heaps
- **For Pears**:
- Number of heaps = Total pears / HCF = 646 / 17 = 38 heaps
- **For Oranges**:
- Number of heaps = Total oranges / HCF = 748 / 17 = 44 heaps

6. **Total Number of Heaps**:
- Total heaps = Heaps of apples + Heaps of pears + Heaps of oranges
- Total heaps = 31 + 38 + 44 = 113 heaps

### Final Answer:
- The greatest number of fruits possible in each heap is **17**.
- The total number of heaps formed is **113**.