The LCM of165, 176, 385 and 495 is k. When is divided by the HCF of the numbers, the quotientis p. What is the value of p?

Options

A. 2520
B. 5040
C. 6720
D. 3360

Answer

To solve the problem, we need to find the LCM (Least Common Multiple) and HCF (Highest Common Factor) of the numbers 165, 176, 385, and 495. Then, we will calculate the quotient \( p \) when the LCM is divided by the HCF.

### Step 1: Find the Prime Factorization of Each Number
1. **165**:
- \( 165 = 3 \times 5 \times 11 \)

2. **176**:
- \( 176 = 2^4 \times 11 \)

3. **385**:
- \( 385 = 5 \times 7 \times 11 \)

4. **495**:
- \( 495 = 3^2 \times 5 \times 11 \)

### Step 2: Calculate the HCF
To find the HCF, we take the lowest power of all prime factors that appear in all numbers.

- The common prime factors are \( 5 \) and \( 11 \).
- The lowest power of \( 5 \) is \( 5^1 \).
- The lowest power of \( 11 \) is \( 11^1 \).

Thus, the HCF is:
\[ HCF = 5^1 \times 11^1 = 5 \times 11 = 55 \]

### Step 3: Calculate the LCM
To find the LCM, we take the highest power of all prime factors that appear in any of the numbers.

- For \( 2 \): highest power is \( 2^4 \) (from 176)
- For \( 3 \): highest power is \( 3^2 \) (from 495)
- For \( 5 \): highest power is \( 5^1 \) (from all)
- For \( 7 \): highest power is \( 7^1 \) (from 385)
- For \( 11 \): highest power is \( 11^1 \) (from all)

Thus, the LCM is:
\[ LCM = 2^4 \times 3^2 \times 5^1 \times 7^1 \times 11^1 \]

Calculating this step-by-step:
- \( 2^4 = 16 \)
- \( 3^2 = 9 \)
- \( 5^1 = 5 \)
- \( 7^1 = 7 \)
- \( 11^1 = 11 \)

Now, calculate the LCM:
\[ LCM = 16 \times 9 \times 5 \times 7 \times 11 \]

Calculating:
1. \( 16 \times 9 = 144 \)
2. \( 144 \times 5 = 720 \)
3. \( 720 \times 7 = 5040 \)
4. \( 5040 \times 11 = 55440 \)

So, \( LCM = 55440 \).

### Step 4: Calculate \( p \)
Now we need to find \( p \) by dividing the LCM by the HCF:
\[ p = \frac{LCM}{HCF} = \frac{55440}{55} \]

Calculating:
\[ p = 1008 \]

### Final Answer
The value of \( p \) is \( 1008 \).

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