Which pair of numbers has an LCM of 18?

Steps to Solve

1. List the factors of 18
   The first step is to identify all the factors of 18. The factors are:
   $1, 2, 3, 6, 9, 18$.

2. Determine pairs of factors
   Next, we need to find pairs of these factors whose greatest common divisor (GCD) is such that the LCM can be calculated directly. Recall the formula for LCM:
   $$ \text{LCM}(a, b) = \frac{a \cdot b}{\text{GCD}(a, b)} $$

3. Calculate LCM for each pair
   We will go through each pair of factors and calculate the LCM using our formula. We will focus on pairs $(a, b)$ where $L = 18$:

   • For pairs $(2, 9)$:

     • $\text{GCD}(2, 9) = 1$
     • $ \text{LCM}(2, 9) = \frac{2 \cdot 9}{1} = 18 $

   • For pairs $(3, 6)$:

     • $\text{GCD}(3, 6) = 3$
     • $ \text{LCM}(3, 6) = \frac{3 \cdot 6}{3} = 6 $ (not valid)

   • For pairs $(6, 9)$:

     • $\text{GCD}(6, 9) = 3$
     • $ \text{LCM}(6, 9) = \frac{6 \cdot 9}{3} = 18 $

   • Repeat this for pairs with $1$ and $18$:

     • $(1, 18)$ gives an LCM of 18
     • Other pairs do not yield 18.

4. List valid pairs
   After calculating, we find the valid pairs that yield an LCM of 18:

• $(2, 9)$
• $(6, 18)$
• $(1, 18)$
• $(9, 2)$ (which is the reverse of $(2, 9)$)