There are 72 green colour, 144 red colour and 234 organe colour chairs in an auditorium. A boy want to put the chairs in rows such that each row is of the same coloured chairs and the number of chairs is equal in each row. Find the minimum number of rows of chairs in the auditorium.

Options

A. 30
B. 25
C. 15
D. 45

Explanation

To solve the problem of arranging chairs in rows such that each row contains the same number of chairs of the same color, we need to find the highest common factor (HCF) of the number of chairs of each color. Here’s how to do it step by step:

### Step 1: Identify the number of chairs of each color
We have:
- Green chairs = 72
- Red chairs = 144
- Orange chairs = 234

### Step 2: Find the HCF of the three numbers
To find the HCF, we will first find the prime factorization of each number.

**Prime Factorization:**
- **72**:
- 72 = 2 × 36
- 36 = 2 × 18
- 18 = 2 × 9
- 9 = 3 × 3
- So, 72 = 2^3 × 3^2

- **144**:
- 144 = 2 × 72
- 72 = 2 × 36
- 36 = 2 × 18
- 18 = 2 × 9
- 9 = 3 × 3
- So, 144 = 2^4 × 3^2

- **234**:
- 234 = 2 × 117
- 117 = 3 × 39
- 39 = 3 × 13
- So, 234 = 2^1 × 3^2 × 13^1

### Step 3: Identify common factors
Now we will identify the common prime factors from the factorizations:
- The common prime factor is 2 and 3.
- The minimum power of 2 in the factorizations is 1 (from 234).
- The minimum power of 3 in the factorizations is 2 (from all three numbers).

### Step 4: Calculate the HCF
Now we calculate the HCF:
HCF = 2^1 × 3^2 = 2 × 9 = 18

### Step 5: Calculate the number of rows
To find the number of rows, we will divide the total number of chairs by the HCF.

**Total number of chairs**:
Total = 72 + 144 + 234 = 450

Now, divide the total number of chairs by the HCF:
Number of rows = Total chairs / HCF = 450 / 18 = 25

### Final Answer
The minimum number of rows of chairs in the auditorium is **25**.

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