Lillian notices that her apartment number is divisible by 22, and her sister notices it is divisible by 44. What is the smallest possible apartment number?

1. Break Down the Problem

To solve this problem, we need to find the smallest apartment number that is divisible by both 22 and 44. The first step is to understand the relationship between these two numbers and find their least common multiple (LCM).

2. Relevant Concepts

The LCM of two numbers can be found using prime factorization or by using the formula: $\text{LCM}(a, b) = \frac{|a \times b|}{\text{GCD}(a, b)}$ For our numbers:

• $a = 22$
• $b = 44$

3. Analysis and Detail

First, let's find the greatest common divisor (GCD) of 22 and 44.

1. The prime factorization of 22 is $2 \times 11$.
2. The prime factorization of 44 is $2^2 \times 11$.

To find the GCD:

• The minimum power of each prime in their factorizations gives us: $\text{GCD}(22, 44) = 2^1 \times 11^1 = 22$

Now, calculate the LCM: $\text{LCM}(22, 44) = \frac{22 \times 44}{22} = 44$

4. Verify and Summarize

We have determined that the smallest apartment number divisible by both 22 and 44 is 44. We can check:

• $44 \div 22 = 2$ (Divisible)
• $44 \div 44 = 1$ (Divisible)

Final Answer

The smallest possible apartment number is $\textbf{44}$.