Given that $a$ and $b$ are two integers, and the conditions are as follows:

1. $ab = x + 16$,
2. The least common multiple (LCM) of $a$ and $b$ is $2x + 8$,
3. The highest common factor (HCF) of $a$ and $b$ is 1.

We need to find the value of the product $ab$.

Step 1: Use the relationship between LCM, HCF, and the product of two numbers

We know that for any two integers $a$ and $b$:

$\text{LCM}(a, b) \times \text{HCF}(a, b) = ab$

Given that $\text{HCF}(a, b) = 1$, the above equation simplifies to:

$\text{LCM}(a, b) = ab$

Step 2: Substitute the given values into the equation

We have:

$\text{LCM}(a, b) = 2x + 8$ $ab = x + 16$

Since $\text{LCM}(a, b) = ab$, we equate the two expressions:

$2x + 8 = x + 16$

Step 3: Solve for $x$

Subtract $x$ from both sides:

$x + 8 = 16$

Subtract 8 from both sides:

$x = 8$

Step 4: Find the value of $ab$

Now that we have $x = 8$, substitute it back into the expression for $ab$:

$ab = x + 16 = 8 + 16 = 24$

So, the value of the product $ab$ is $\boxed{24}$.

Would you like more details, or do you have any further questions?

Related Questions:

1. How do you determine the LCM and HCF of two integers?
2. What is the significance of HCF being 1 in this problem?
3. How would the solution change if HCF(a, b) was greater than 1?
4. Can this method be used for non-integer values of $a$ and $b$?
5. How does the relationship between LCM, HCF, and the product apply in other contexts?

Tip: Always check whether LCM and HCF conditions are consistent with the product of the numbers to verify correctness.