Let a, b be two integers. If ab = x + 16, LCM of a, b is 2x + 8 and HCF of a, b is 1 then
find the value of product ab.

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.

Let a, b be two integers. If ab = x + 16, LCM of a, b is 2x + 8 and HCF of a, b is 1, then find the value of product ab.

Explore how to find the product of two integers a and b when given that their product equals x + 16, their LCM is 2x + 8, and their HCF is 1. This problem involves key concepts in number theory and basic algebra, ideal for students in Grades 9-10.

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