Find the largest number which when substracted from 5834 then the obtained number will be divisible by 20, 28, 32 and 35 ?

Options

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

Answer

To find the largest number which, when subtracted from 5834, results in a number that is divisible by 20, 28, 32, and 35, we will follow these steps:

### Step 1: Find the LCM of the numbers 20, 28, 32, and 35.
To find the LCM, we first need to factor each number into its prime factors:

- **20** = 2^2 × 5
- **28** = 2^2 × 7
- **32** = 2^5
- **35** = 5 × 7

### Step 2: Identify the highest power of each prime factor.
Now, we take the highest power of each prime factor from the factorizations:

- For **2**, the highest power is 2^5 (from 32).
- For **5**, the highest power is 5^1 (from both 20 and 35).
- For **7**, the highest power is 7^1 (from both 28 and 35).

### Step 3: Calculate the LCM.
Now we multiply these highest powers together to get the LCM:

\[ \text{LCM} = 2^5 × 5^1 × 7^1 = 32 × 5 × 7 \]

Calculating this step-by-step:

1. \( 32 × 5 = 160 \)
2. \( 160 × 7 = 1120 \)

So, the LCM of 20, 28, 32, and 35 is **1120**.

### Step 4: Subtract the LCM from 5834.
Now, we need to subtract the LCM from 5834 to find the largest number that can be subtracted:

\[ 5834 - 1120 = 4714 \]

### Conclusion:
The largest number which, when subtracted from 5834, results in a number that is divisible by 20, 28, 32, and 35 is **4714**.

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