We are given the following conditions:

• The product of positive numbers $A$ and $B$ is 1008, i.e., $A \times B = 1008$
• $A$ is 7 times $B$, i.e., $A = 7B$

Step 1: Substitute $A = 7B$ into the product equation.

Substituting $A = 7B$ into the equation $A \times B = 1008$: $7B \times B = 1008$ $7B^2 = 1008$

Step 2: Solve for $B$.

Now, divide both sides of the equation by 7 to solve for $B^2$: $B^2 = \frac{1008}{7}$ $B^2 = 144$

Taking the square root of both sides: $B = \sqrt{144} = 12$

Step 3: Find $A$.

Now that we know $B = 12$, we can substitute this into the equation $A = 7B$ to find $A$: $A = 7 \times 12 = 84$

Thus, the value of $A$ is $\boxed{84}$.

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Would you like further details on this solution or have any questions? Here are some related questions:

1. What if the relationship between $A$ and $B$ was different (e.g., $A = 5B$)? How would you solve it?
2. How can we check that the values of $A$ and $B$ satisfy the original product equation?
3. What happens if one of the numbers, say $A$, is negative? How would the solution change?
4. Can this type of problem be extended to more than two variables?
5. How would you solve this problem if we were also given that the sum $A + B = 100$?

Tip: Always substitute given relationships into the equation first, as this can simplify solving the problem significantly.