The smallest number which when diminished by 3 is divisible by 14, 28, 36 and 45 is

Options

A. 1257
B. 1260
C. 1263
D. none of these

Answer

To find the smallest number which, when diminished by 3, is divisible by 14, 28, 36, and 45, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the numbers**: We need to find the smallest number that, when reduced by 3, is divisible by 14, 28, 36, and 45.

2. **Calculate the LCM**: We need to find the Least Common Multiple (LCM) of the numbers 14, 28, 36, and 45.
- First, we find the prime factorization of each number:
- 14 = 2 × 7
- 28 = 2² × 7
- 36 = 2² × 3²
- 45 = 3² × 5

3. **Determine the LCM**: The LCM is found by taking the highest power of each prime number from the factorizations:
- From 14: 2¹ and 7¹
- From 28: 2² (highest power) and 7¹
- From 36: 2² and 3² (highest power)
- From 45: 3² and 5¹

Therefore, the LCM is:
- LCM = 2² × 3² × 5¹ × 7¹

4. **Calculate the LCM**:
- LCM = 4 × 9 × 5 × 7
- First, calculate 4 × 9 = 36
- Then, 36 × 5 = 180
- Finally, 180 × 7 = 1260

So, the LCM of 14, 28, 36, and 45 is 1260.

5. **Find the required number**: Since we need a number that is divisible by this LCM when diminished by 3, we set up the equation:
- Let the required number be \( x \).
- Thus, \( x - 3 = 1260 \)
- Therefore, \( x = 1260 + 3 = 1263 \)

### Final Answer:
The smallest number which when diminished by 3 is divisible by 14, 28, 36, and 45 is **1263**.

---