Some booklets are packed in a packing unit.Shinde said that if he packed 50 booklets in a bundle, no booklet left behind. Ramdas saw that if 70 booklets are packed in a bundle, no booklet left behind. Beeju said that if 20 booklets are packed in a bundle, no booklet left behind. If all are speaking true, find the minimum booklets.

Options

A. A)540
B. B)660
C. C)670
D. D)700

Answer

To find the minimum number of booklets that satisfy the conditions given by Shinde, Ramdas, and Beeju, we need to determine the least common multiple (LCM) of the numbers 50, 70, and 20. Here’s how we can do that step by step:

### Step 1: Prime Factorization
First, we will find the prime factorization of each number:
- **50**:
- 50 = 2 × 5²
- **70**:
- 70 = 2 × 5 × 7
- **20**:
- 20 = 2² × 5

### Step 2: Identify the Highest Powers of Each Prime
Next, we identify the highest power of each prime number that appears in the factorizations:
- For **2**, the highest power is \(2²\) (from 20).
- For **5**, the highest power is \(5²\) (from 50).
- For **7**, the highest power is \(7¹\) (from 70).

### Step 3: Calculate the LCM
Now, we can calculate the LCM by multiplying these highest powers together:
\[ \text{LCM} = 2² × 5² × 7¹ \]
Calculating this step by step:
- \(2² = 4\)
- \(5² = 25\)
- \(7¹ = 7\)

Now, multiply these results:
\[ 4 × 25 = 100 \]
\[ 100 × 7 = 700 \]

### Step 4: Conclusion
Thus, the minimum number of booklets that can be packed without any left behind is **700**.