To determine which number completes the statement "The least common multiple of ___, 12, and 18 is 36," we need to check each given option to see if its least common multiple (LCM) with 12 and 18 gives us 36.
We know that the LCM of two numbers, \(a\) and \(b\), is the smallest positive integer that is divisible by both \(a\) and \(b\). The general formula for finding the LCM of two numbers is:
[tex]\[ \text{LCM}(a, b) = \frac{|a \times b|}{\text{GCD}(a, b)} \][/tex]
where GCD is the greatest common divisor.
Given:
- The numbers 12 and 18
First, we'll find the LCM of 12 and 18:
[tex]\[ \text{LCM}(12, 18) \][/tex]
Steps:
1. Find the prime factorization of each number:
- \(12 = 2^2 \times 3\)
- \(18 = 2 \times 3^2\)
2. For the LCM, take the highest power of each prime that appears in the factorizations:
- \( \text{LCM} = 2^2 \times 3^2 = 4 \times 9 = 36 \)
Now we know that the LCM of 12 and 18 is 36. To satisfy the condition \( \text{LCM}(x, \text{LCM}(12, 18)) = 36 \), we can simplify it to \( \text{LCM}(x, 36) = 36 \).
We need to identify the number from the options (5, 7, 8, 9) that, when used to find the LCM with 36, results in 36 itself.
Testing each option:
1. \( \text{LCM}(36, 5) = 180 \):
[tex]\[
\begin{aligned}
\text{Prime factors: } & 36 = 2^2 \times 3^2 \\
& 5 = 5^1 \\
\text{LCM} \rightarrow & 2^2 \times 3^2 \times 5 = 180 \\
\end{aligned}
\][/tex]
This is not 36.
2. \( \text{LCM}(36, 7) = 252 \):
[tex]\[
\begin{aligned}
\text{Prime factors: } & 36 = 2^2 \times 3^2 \\
& 7 = 7^1 \\
\text{LCM} \rightarrow & 2^2 \times 3^2 \times 7 = 252 \\
\end{aligned}
\][/tex]
This is not 36.
3. \( \text{LCM}(36, 8) = 72 \):
[tex]\[
\begin{aligned}
\text{Prime factors: } & 36 = 2^2 \times 3^2 \\
& 8 = 2^3 \\
\text{LCM} \rightarrow & 2^3 \times 3^2 = 72 \\
\end{aligned}
\][/tex]
This is not 36.
4. \( \text{LCM}(36, 9) = 36 \):
[tex]\[
\begin{aligned}
\text{Prime factors: } & 36 = 2^2 \times 3^2 \\
& 9 = 3^2 \\
\text{LCM} \rightarrow & 2^2 \times 3^2 = 36 \\
\end{aligned}
\][/tex]
This correctly results in 36.
Therefore, the number that completes the statement "The least common multiple of ___, 12, and 18 is 36" is:
[tex]\[ \boxed{9} \][/tex]
