Three brands of pens A, B and Care available in packets of 10, 12 and 24 respectively. If a shopkeeper wants to buy equal number of pens of each brand, what is the minimum number of packets of each brand, he should buy? 

Options

A. A = 10, B = 12, C = 5
B. A = 5, B = 12, C = 10
C. A = 10, B = 5C = 12
D. A = 12, B = 10, C = 5

Answer

To find the minimum number of packets of each brand (A, B, and C) that the shopkeeper should buy to have an equal number of pens, we will follow these steps:

### Step 1: Identify the number of pens in each packet
- Brand A has 10 pens per packet.
- Brand B has 12 pens per packet.
- Brand C has 24 pens per packet.

### Step 2: Find the Least Common Multiple (LCM) of the packet sizes
To find the minimum number of pens that can be bought equally from all three brands, we need to calculate the LCM of 10, 12, and 24.

- The prime factorization of each number is:
- 10 = 2 × 5
- 12 = 2² × 3
- 24 = 2³ × 3

- The LCM is found by taking the highest power of each prime number:
- For 2: the highest power is 2³ (from 24)
- For 3: the highest power is 3¹ (from 12 and 24)
- For 5: the highest power is 5¹ (from 10)

Thus, the LCM = 2³ × 3¹ × 5¹ = 8 × 3 × 5 = 120.

### Step 3: Calculate the number of packets needed for each brand
Now that we have the LCM (120), we can determine how many packets of each brand are needed to reach this total.

- For Brand A:
\[ \text{Number of packets} = \frac{120}{10} = 12 \text{ packets} \]

- For Brand B:
\[ \text{Number of packets} = \frac{120}{12} = 10 \text{ packets} \]

- For Brand C:
\[ \text{Number of packets} = \frac{120}{24} = 5 \text{ packets} \]

### Conclusion
The shopkeeper should buy:
- 12 packets of Brand A,
- 10 packets of Brand B,
- 5 packets of Brand C.