The greatest number, which when subtracted from 5834, gives a number exactly divisible by each of 20, 28, 32 and 35 is

Options

A. 1120
B. 4714
C. 5200
D. 5600

Answer

To find the greatest number which, when subtracted from 5834, gives a result that is exactly divisible by 20, 28, 32, and 35, we can follow these steps:

### Step-by-Step Solution:

1. **Find the LCM of the Numbers**:
- We need to find the least common multiple (LCM) of the numbers 20, 28, 32, and 35.
- First, we will find the prime factorization of each number:
- \(20 = 2^2 \times 5^1\)
- \(28 = 2^2 \times 7^1\)
- \(32 = 2^5\)
- \(35 = 5^1 \times 7^1\)

2. **Determine the Highest Powers of Each Prime**:
- From the factorizations, we take the highest power of each prime:
- For \(2\), the highest power is \(2^5\) (from 32).
- For \(5\), the highest power is \(5^1\) (from 20 and 35).
- For \(7\), the highest power is \(7^1\) (from 28 and 35).

3. **Calculate the LCM**:
- Now, we multiply these highest powers together to find the LCM:
\[ \text{LCM} = 2^5 \times 5^1 \times 7^1 = 32 \times 5 \times 7 \]
- First, calculate \(32 \times 5 = 160\).
- Then, calculate \(160 \times 7 = 1120\).
- Thus, the LCM of 20, 28, 32, and 35 is 1120.

4. **Subtract the LCM from 5834**:
- To find the greatest number that can be subtracted from 5834, we need to subtract the LCM (1120) from 5834:
\[ 5834 - 1120 = 4714 \]

5. **Conclusion**:
- The greatest number which when subtracted from 5834 gives a number exactly divisible by 20, 28, 32, and 35 is **4714**.