Find the least number greater than 10 which when divided by 15,20 and25 leaves remainder 10 in each case.

Options

A. 290
B. 300
C. 310
D. 320

Answer

To find the least number greater than 10 which, when divided by 15, 20, and 25, leaves a remainder of 10 in each case, we can follow these steps:

### Step 1: Understand the Problem
We need to find a number \( N \) such that:
- \( N \mod 15 = 10 \)
- \( N \mod 20 = 10 \)
- \( N \mod 25 = 10 \)

This means that \( N - 10 \) should be divisible by 15, 20, and 25.

### Step 2: Set Up the Equation
Let \( M = N - 10 \). Then, we need to find \( M \) such that:
- \( M \mod 15 = 0 \)
- \( M \mod 20 = 0 \)
- \( M \mod 25 = 0 \)

This implies that \( M \) must be a common multiple of 15, 20, and 25.

### Step 3: Find the Least Common Multiple (LCM)
To find \( M \), we need to calculate the LCM of 15, 20, and 25.

1. **Prime Factorization**:
- \( 15 = 3 \times 5 \)
- \( 20 = 2^2 \times 5 \)
- \( 25 = 5^2 \)

2. **Determine the LCM**:
- Take the highest power of each prime factor:
- \( 2^2 \) from 20
- \( 3^1 \) from 15
- \( 5^2 \) from 25

Therefore, the LCM is:
\[ LCM = 2^2 \times 3^1 \times 5^2 = 4 \times 3 \times 25 \]

3. **Calculate the LCM**:
\[ 4 \times 3 = 12 \]
\[ 12 \times 25 = 300 \]

Thus, \( M = 300 \).

### Step 4: Find \( N \)
Now, since \( M = N - 10 \):
\[ N = M + 10 = 300 + 10 = 310 \]

### Step 5: Conclusion
The least number greater than 10 which, when divided by 15, 20, and 25, leaves a remainder of 10 is **310**.

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