What is the smallest number which when divided by 8,12 and 14 gives the remainder 6 ?

Options

A. 174
B. 168
C. 162
D. 154

Answer

To find the smallest number that, when divided by 8, 12, and 14, gives a remainder of 6, we can follow these steps:

### Step 1: Find the LCM of 8, 12, and 14
To find the least common multiple (LCM), we first need to determine the prime factorization of each number:

- **8** = \(2^3\)
- **12** = \(2^2 \times 3^1\)
- **14** = \(2^1 \times 7^1\)

Now, we take the highest power of each prime factor:

- For \(2\): the highest power is \(2^3\) (from 8)
- For \(3\): the highest power is \(3^1\) (from 12)
- For \(7\): the highest power is \(7^1\) (from 14)

Thus, the LCM is calculated as follows:
\[ \text{LCM} = 2^3 \times 3^1 \times 7^1 = 8 \times 3 \times 7 \]

Calculating this gives:
\[ 8 \times 3 = 24 \]
\[ 24 \times 7 = 168 \]

So, the LCM of 8, 12, and 14 is **168**.

### Step 2: Adjust for the Remainder
Since we need a number that gives a remainder of 6 when divided by 8, 12, and 14, we can express our desired number \(N\) in terms of the LCM:
\[ N = \text{LCM} + 6 \]
Substituting the LCM we found:
\[ N = 168 + 6 = 174 \]

### Step 3: Verify the Result
To ensure that 174 gives a remainder of 6 when divided by 8, 12, and 14, we can check each division:

- **174 ÷ 8**:
- \(174 = 8 \times 21 + 6\) (Remainder is 6)

- **174 ÷ 12**:
- \(174 = 12 \times 14 + 6\) (Remainder is 6)

- **174 ÷ 14**:
- \(174 = 14 \times 12 + 6\) (Remainder is 6)

Since all divisions yield a remainder of 6, we confirm that the smallest number is **174**.

### Final Answer
The smallest number which when divided by 8, 12, and 14 gives a remainder of 6 is **174**.

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