The least number which should be subtracted from 1936 so that the remainder, which when divided by 9, 10 and 15 leaves the remainder 7 in each cases is

Options

A. 37
B. 36
C. 39
D. 30

Answer

To solve the problem step by step, we need to find the least number that should be subtracted from 1936 so that the remainder, when divided by 9, 10, and 15, leaves a remainder of 7 in each case.

### Step 1: Find the LCM of 9, 10, and 15
To ensure that the remainder is 7 when divided by 9, 10, and 15, we first need to find the least common multiple (LCM) of these numbers.

- **Prime factorization:**
- 9 = \(3^2\)
- 10 = \(2^1 \times 5^1\)
- 15 = \(3^1 \times 5^1\)

- **LCM Calculation:**
- Take the highest power of each prime factor:
- \(2^1\) from 10
- \(3^2\) from 9
- \(5^1\) from both 10 and 15
- Therefore, LCM = \(2^1 \times 3^2 \times 5^1 = 2 \times 9 \times 5 = 90\)

### Step 2: Determine the number to be subtracted
Now, we need to find the largest multiple of 90 that is less than or equal to 1936.

- **Divide 1936 by 90:**
\[ 1936 \div 90 \approx 21.0222 \]
- The largest integer less than or equal to 21.0222 is 21.

- **Calculate the largest multiple of 90:**
\[ 21 \times 90 = 1890 \]

### Step 3: Calculate the remainder
Now, we need to find the difference between 1936 and this multiple of 90 (1890).

- **Subtract to find the difference:**
\[ 1936 - 1890 = 46 \]

### Step 4: Adjust for the required remainder
Since we want the remainder to be 7 when divided by 9, 10, and 15, we need to add 7 to the largest multiple of 90.

- **Add 7 to 1890:**
\[ 1890 + 7 = 1897 \]

### Step 5: Final calculation
Now, we need to find how much we subtracted from 1936 to get to 1897.

- **Final subtraction:**
\[ 1936 - 1897 = 39 \]

### Conclusion
The least number that should be subtracted from 1936 is **39**.

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