The least number of 5-digits which exactly divided by 52, 56, 78 and 91 ?

Options

A. 10290
B. 10860
C. 10920
D. 10580

Answer

To find the least number of 5 digits that is exactly divisible by 52, 56, 78, and 91, we will follow these steps:

### Step 1: Find the LCM of the given numbers
To find the least common multiple (LCM) of 52, 56, 78, and 91, we first need to perform the prime factorization of each number.

- **52**:
- Prime factorization: \(52 = 2^2 \times 13^1\)

- **56**:
- Prime factorization: \(56 = 2^3 \times 7^1\)

- **78**:
- Prime factorization: \(78 = 2^1 \times 3^1 \times 13^1\)

- **91**:
- Prime factorization: \(91 = 7^1 \times 13^1\)

### Step 2: Determine the highest power of each prime factor
Now we will take the highest power of each prime factor from the factorizations:

- For \(2\): The highest power is \(2^3\) (from 56).
- For \(3\): The highest power is \(3^1\) (from 78).
- For \(7\): The highest power is \(7^1\) (from 56 and 91).
- For \(13\): The highest power is \(13^1\) (from 52, 78, and 91).

### Step 3: Calculate the LCM
Now we can calculate the LCM by multiplying these highest powers together:

\[ \text{LCM} = 2^3 \times 3^1 \times 7^1 \times 13^1 \]

Calculating this step-by-step:

1. \(2^3 = 8\)
2. \(3^1 = 3\)
3. \(7^1 = 7\)
4. \(13^1 = 13\)

Now multiply these together:

\[ 8 \times 3 = 24 \]
\[ 24 \times 7 = 168 \]
\[ 168 \times 13 = 2184 \]

So, the LCM of 52, 56, 78, and 91 is **2184**.

### Step 4: Find the least 5-digit number divisible by the LCM
The smallest 5-digit number is **10000**. We need to find the smallest multiple of 2184 that is greater than or equal to 10000.

To do this, we divide 10000 by 2184 and round up to the nearest whole number:

\[ \frac{10000}{2184} \approx 4.577 \]

Rounding up gives us **5**.

### Step 5: Calculate the least 5-digit number
Now we multiply 2184 by 5 to find the least 5-digit number:

\[ 2184 \times 5 = 10920 \]

Thus, the least number of 5 digits which is exactly divisible by 52, 56, 78, and 91 is **10920**.

### Summary of Steps:
1. Find the prime factorization of each number.
2. Determine the highest power of each prime factor.
3. Calculate the LCM using these highest powers.
4. Find the smallest multiple of the LCM that is a 5-digit number.