Let `x` be the least number between 70000 and 75000 which on being divided by 225, 250 and 275 leaves a remainder of 61 in each case. The sum of the digits of `x` is :

Options

A. 12
B. 29
C. 16
D. 21

Answer

To solve the problem, we need to find the least number \( x \) between 70,000 and 75,000 that, when divided by 225, 250, and 275, leaves a remainder of 61. Here’s how we can approach the solution step-by-step:

### Step 1: Find the LCM of 225, 250, and 275
To find the least common multiple (LCM), we need to factor each number into its prime factors:

- \( 225 = 3^2 \times 5^2 \)
- \( 250 = 2 \times 5^3 \)
- \( 275 = 5^2 \times 11 \)

Now, we take the highest power of each prime factor:

- For \( 2 \): \( 2^1 \) (from 250)
- For \( 3 \): \( 3^2 \) (from 225)
- For \( 5 \): \( 5^3 \) (from 250)
- For \( 11 \): \( 11^1 \) (from 275)

Thus, the LCM is:
\[ LCM = 2^1 \times 3^2 \times 5^3 \times 11^1 \]

Calculating this step-by-step:
- \( 5^3 = 125 \)
- \( 3^2 = 9 \)
- \( 2^1 = 2 \)
- \( 11^1 = 11 \)

Now, calculate:
\[ LCM = 2 \times 9 \times 125 \times 11 \]

Calculating \( 2 \times 9 = 18 \):
\[ 18 \times 125 = 2250 \]
Now, calculate \( 2250 \times 11 = 24750 \).

### Step 2: Find multiples of the LCM between 70,000 and 75,000
Next, we need to find multiples of 24750 that fall between 70,000 and 75,000.

Calculating the multiples:
- \( 24750 \times 2 = 49500 \) (too low)
- \( 24750 \times 3 = 74250 \) (within range)

### Step 3: Adjust for the remainder
Since we need \( x \) to leave a remainder of 61 when divided by 225, 250, and 275, we can find \( x \) as follows:
\[ x = 74250 + 61 = 74311 \]

### Step 4: Sum the digits of \( x \)
Now, we need to find the sum of the digits of \( 74311 \):
\[ 7 + 4 + 3 + 1 + 1 = 16 \]

### Final Answer
The sum of the digits of \( x \) is \( \boxed{16} \).