Evaluate the least number which when divided by the numbers 18, 24, 30 and 42 leaves a remainder of 1.

Options

A. 4221
B. 2521
C. 3862
D. 1221

Answer

To find the least number which, when divided by 18, 24, 30, and 42, leaves a remainder of 1, we can follow these steps:

### Step 1: Find the Least Common Multiple (LCM) of the numbers
We need to find the LCM of 18, 24, 30, and 42.

- **Prime Factorization**:
- 18 = 2 × 3²
- 24 = 2³ × 3
- 30 = 2 × 3 × 5
- 42 = 2 × 3 × 7

- **Taking the highest power of each prime**:
- For 2: highest power is 2³ (from 24)
- For 3: highest power is 3² (from 18)
- For 5: highest power is 5¹ (from 30)
- For 7: highest power is 7¹ (from 42)

Thus, the LCM is:
\[ LCM = 2³ × 3² × 5¹ × 7¹ \]

### Step 2: Calculate the LCM
Now, we calculate the LCM:
\[ LCM = 8 × 9 × 5 × 7 \]

Calculating step by step:
- First, calculate \(8 × 9 = 72\)
- Then, calculate \(72 × 5 = 360\)
- Finally, calculate \(360 × 7 = 2520\)

So, \(LCM(18, 24, 30, 42) = 2520\).

### Step 3: Find the least number that leaves a remainder of 1
Since we need a number that leaves a remainder of 1 when divided by these numbers, we can express this as:
\[ N = LCM + 1 = 2520 + 1 = 2521 \]

### Conclusion
The least number which when divided by 18, 24, 30, and 42 leaves a remainder of 1 is:
\[ \boxed{2521} \]