Question

Kelsey notices that her apartment number is divisible by 10, and her sister notices it is divisible by 9. What is the smallest possible apartment number?

Answer 1

To find the smallest possible apartment number Kelsey can have, we need to determine the smallest number that is divisible by both 10 and 9.

1. **Divisibility by 10**: A number is divisible by 10 if it ends in 0. Thus, we can express any number divisible by 10 in the form:
\[
n = 10k \quad \text{for some integer } k.
\]

2. **Divisibility by 9**: A number is divisible by 9 if the sum of its digits is divisible by 9.

Next, we will find the least common multiple (LCM) of 10 and 9 to find the smallest number that meets both divisibility conditions.

3. **Finding the LCM**:
- The prime factorizations are:
\[
10 = 2 \times 5,
\]
\[
9 = 3^2.
\]
- To find the LCM, we take each prime factor at its highest power in these factorizations:
\[
\text{lcm}(10, 9) = 2^1 \times 3^2 \times 5^1 = 2 \times 9 \times 5.
\]

4. **Calculating the LCM**:
\[
2 \times 9 = 18,
\]
\[
18 \times 5 = 90.
\]
Thus, \(\text{lcm}(10, 9) = 90\).

The least common multiple is 90, which means that 90 is the smallest number divisible by both 10 and 9.

5. **Verification**:
- Check if 90 is divisible by 10:
\[
90 \div 10 = 9 \quad \text{(which is an integer)}.
\]
- Check if 90 is divisible by 9:
\[
90 \div 9 = 10 \quad \text{(which is also an integer)}.
\]

Since both conditions are satisfied, the smallest possible apartment number Kelsey can have is

\[
\boxed{90}.
\]