The smallest number which when divided by 30, 45, 75 and 60 leaves a remainder of 21, 36, 66 and 51 respectively is

Options

A. `900`
B. `909`
C. `891`
D. none of these

Answer

To solve the problem step by step, we need to find the smallest number that leaves specific remainders when divided by given divisors. Here’s how to approach it:

### Step 1: Identify the Divisors and Remainders
We are given the following divisors and their corresponding remainders:
- Divisor 1: 30, Remainder 1: 21
- Divisor 2: 45, Remainder 2: 36
- Divisor 3: 75, Remainder 3: 66
- Divisor 4: 60, Remainder 4: 51

### Step 2: Calculate the Differences
Next, we calculate the difference between each divisor and its corresponding remainder:
- For 30 and 21: \(30 - 21 = 9\)
- For 45 and 36: \(45 - 36 = 9\)
- For 75 and 66: \(75 - 66 = 9\)
- For 60 and 51: \(60 - 51 = 9\)

### Step 3: Confirm the Common Difference
All the differences calculated are the same, which is 9. This means that the number we are looking for can be expressed in the form of \(N = k \cdot \text{LCM} - 9\), where \(k\) is an integer and LCM is the least common multiple of the divisors.

### Step 4: Calculate the LCM of the Divisors
Now, we need to find the LCM of the divisors: 30, 45, 75, and 60.

1. **Prime Factorization**:
- \(30 = 2 \times 3 \times 5\)
- \(45 = 3^2 \times 5\)
- \(75 = 3 \times 5^2\)
- \(60 = 2^2 \times 3 \times 5\)

2. **Identify the highest powers of all prime factors**:
- For \(2\): highest power is \(2^2\) (from 60)
- For \(3\): highest power is \(3^2\) (from 45)
- For \(5\): highest power is \(5^2\) (from 75)

3. **Calculate the LCM**:
\[ \text{LCM} = 2^2 \times 3^2 \times 5^2 = 4 \times 9 \times 25 = 900 \]

### Step 5: Calculate the Required Number
Now, we can find the smallest number \(N\):
\[ N = \text{LCM} - 9 = 900 - 9 = 891 \]

### Final Answer
The smallest number which when divided by 30, 45, 75, and 60 leaves a remainder of 21, 36, 66, and 51 respectively is **891**.

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