Find the product of the least common multiple (LCM) of $8$ and $6$ and the greatest common divisor (GCD) of $8$ and $6$.
Since $6 = 2 \cdot 3$ and $3$ is relatively prime with $8$ while $2$ divides into both $6$ and $8$, it follows that $\text{gcd}(6,8) = 2$. The multiples of $8$ are $8, 16, 24, 32\ldots$ and the multiples of $6$ are $6, 12, 18, 24, 30, \ldots$, so $\text{lcm}(6,8) = 24$. Thus, the desired product is $2 \times 24 = \boxed{48}.$

Notice that this product is equal to the product of the original numbers, $6\times 8$. Is there a reason for that?
48
Level 2
Prealgebra
Problem: Find the product of the least common multiple (LCM) of $8$ and $6$ and the greatest common divisor (GCD) of $8$ and $6$.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the product of the LCM and GCD of two numbers, we can use the relationship between these concepts. The formula states that for any two integers \( a \) and \( b \):

\[
\text{LCM}(a, b) \times \text{GCD}(a, b) = a \times b
\]

In this case, let \( a = 8 \) and \( b = 6 \). First, we need to calculate the GCD of \( 8 \) and \( 6 \). The GCD is the largest integer that divides both numbers without leaving a remainder. The divisors of \( 8 \) are \( 1, 2, 4, 8 \) and the divisors of \( 6 \) are \( 1, 2, 3, 6 \). The common divisors are \( 1 \) and \( 2 \), so:

\[
\text{GCD}(8, 6) = 2
\]

Next, we calculate the LCM of \( 8 \) and \( 6 \). The LCM is the smallest integer that is a multiple of both numbers. The multiples of \( 8 \) are \( 8, 16, 24, 32, \ldots \) and the multiples of \( 6 \) are \( 6, 12, 18, 24, 30, \ldots \). The smallest common multiple is \( 24 \), thus:

\[
\text{LCM}(8, 6) = 24
\]

Now, we can apply the relationship between LCM and GCD:

\[
\text{LCM}(8, 6) \times \text{GCD}(8, 6) = 24 \times 2
\]

Calculating this gives:

\[
24 \times 2 = 48
\]

Therefore, the final answer is \( 48 \).<|END_OF_LATENT|> The final answer is $48$.