Determine the least number which when divided by 90, 108 and 126 leaves a remainder 6.

Options

A. 3774
B. 3786
C. 3788
D. 3780

Answer

To determine the least number which, when divided by 90, 108, and 126, leaves a remainder of 6, we can follow these steps:

### Step 1: Find the LCM of 90, 108, and 126
To find the least common multiple (LCM), we first need to factor each number into its prime factors.

- **Prime factorization of 90:**
- 90 = 2 × 3² × 5

- **Prime factorization of 108:**
- 108 = 2² × 3³

- **Prime factorization of 126:**
- 126 = 2 × 3² × 7

### Step 2: Identify the highest power of each prime factor
Now, we will take the highest power of each prime factor from the factorizations:

- For the prime factor **2**, the highest power is \(2²\) (from 108).
- For the prime factor **3**, the highest power is \(3³\) (from 108).
- For the prime factor **5**, the highest power is \(5¹\) (from 90).
- For the prime factor **7**, the highest power is \(7¹\) (from 126).

### Step 3: Calculate the LCM
Now, we can calculate the LCM by multiplying these highest powers together:

\[ \text{LCM} = 2² × 3³ × 5¹ × 7¹ \]

Calculating this step-by-step:

- \(2² = 4\)
- \(3³ = 27\)
- \(5¹ = 5\)
- \(7¹ = 7\)

Now, multiply these values together:

\[ \text{LCM} = 4 × 27 × 5 × 7 \]

Calculating it step by step:

1. \(4 × 27 = 108\)
2. \(108 × 5 = 540\)
3. \(540 × 7 = 3780\)

Thus, the LCM of 90, 108, and 126 is **3780**.

### Step 4: Adjust for the remainder
The problem states that the number leaves a remainder of 6 when divided by 90, 108, and 126. Therefore, we need to add this remainder to the LCM:

\[ \text{Least number} = \text{LCM} + 6 = 3780 + 6 = 3786 \]

### Final Answer
The least number which, when divided by 90, 108, and 126, leaves a remainder of 6 is **3786**.

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