Miss Sumitha has three drumsticks in her kitchen which are 72 cm ,90 cm and 84 cm, respectively. She wanted to cut them into equal size of pieces. Find the minimum number of pieces that she can cut.

Options

A. 41
B. 58
C. 48
D. 54

Explanation

To solve the problem of how to cut the drumsticks into equal-sized pieces, we need to find the greatest common divisor (GCD) of the lengths of the drumsticks. The lengths are 72 cm, 90 cm, and 84 cm.

### Step 1: Find the prime factorization of each length.

1. **For 72 cm:**
- 72 = 2 × 36
- 36 = 2 × 18
- 18 = 2 × 9
- 9 = 3 × 3
- So, the prime factorization of 72 is: **2^3 × 3^2**

2. **For 90 cm:**
- 90 = 2 × 45
- 45 = 3 × 15
- 15 = 3 × 5
- So, the prime factorization of 90 is: **2^1 × 3^2 × 5^1**

3. **For 84 cm:**
- 84 = 2 × 42
- 42 = 2 × 21
- 21 = 3 × 7
- So, the prime factorization of 84 is: **2^2 × 3^1 × 7^1**

### Step 2: Identify the common prime factors.

Now we will look for the common prime factors in all three numbers:

- The common prime factor is **2** and **3**.

### Step 3: Find the minimum power of each common prime factor.

- For **2**:
- In 72, the power is 3 (2^3).
- In 90, the power is 1 (2^1).
- In 84, the power is 2 (2^2).
- The minimum power is **1**.

- For **3**:
- In 72, the power is 2 (3^2).
- In 90, the power is 2 (3^2).
- In 84, the power is 1 (3^1).
- The minimum power is **1**.

### Step 4: Calculate the GCD.

Now we can calculate the GCD using the minimum powers of the common prime factors:

GCD = 2^1 × 3^1 = 2 × 3 = **6 cm**.

### Step 5: Calculate the number of pieces.

Now, we can find the number of pieces each drumstick can be cut into:

- For 72 cm: 72 ÷ 6 = 12 pieces.
- For 90 cm: 90 ÷ 6 = 15 pieces.
- For 84 cm: 84 ÷ 6 = 14 pieces.

### Step 6: Find the total number of pieces.

Now, we add the number of pieces:

Total pieces = 12 + 15 + 14 = **41 pieces**.

### Final Answer:

Miss Sumitha can cut the drumsticks into a total of **41 pieces** of 6 cm each.

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