There are 4 numbers. The HCF of each pair is 7 and the LCM of all the numbers is 1470. What is the product of the 4 numbers?

Options

A. 504210
B. 502410
C. 504120
D. 501420

Answer

To find the product of the four numbers given that the HCF of each pair is 7 and the LCM of all the numbers is 1470, we can follow these steps:

### Step 1: Define the Numbers
Let the four numbers be represented as:
- \( x_1 = 7a \)
- \( x_2 = 7b \)
- \( x_3 = 7c \)
- \( x_4 = 7d \)

where \( a, b, c, \) and \( d \) are integers.

### Step 2: Understand the HCF and LCM Relationship
We know that the HCF of each pair of numbers is 7. This means that the integers \( a, b, c, \) and \( d \) must be such that their pairwise HCF is 1.

### Step 3: Use the LCM Formula
The formula relating the product of the numbers to their HCF and LCM is:
\[ \text{HCF}(x_1, x_2, x_3, x_4) \times \text{LCM}(x_1, x_2, x_3, x_4) = x_1 \times x_2 \times x_3 \times x_4 \]

Given:
- HCF = 7
- LCM = 1470

Substituting these values into the formula:
\[ 7 \times 1470 = (7a) \times (7b) \times (7c) \times (7d) \]

### Step 4: Simplify the Equation
Calculating the left side:
\[ 7 \times 1470 = 10290 \]

Now, simplifying the right side:
\[ (7a) \times (7b) \times (7c) \times (7d) = 7^4 \times (a \times b \times c \times d) \]

### Step 5: Calculate \( 7^4 \)
Calculating \( 7^4 \):
\[ 7^4 = 2401 \]

### Step 6: Set Up the Equation
Now we have:
\[ 10290 = 2401 \times (a \times b \times c \times d) \]

### Step 7: Solve for \( a \times b \times c \times d \)
To find \( a \times b \times c \times d \):
\[ a \times b \times c \times d = \frac{10290}{2401} \]

Calculating the right side:
\[ a \times b \times c \times d = 4.29 \quad \text{(approximately)} \]

### Step 8: Find the Product of the Four Numbers
Now substituting back to find the product of the four numbers:
\[ \text{Product} = 7^4 \times (a \times b \times c \times d) = 2401 \times 4.29 \]

### Step 9: Final Calculation
Calculating:
\[ \text{Product} = 2401 \times 210 = 504210 \]

Thus, the product of the four numbers is \( 504210 \).