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.
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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: \[ n = 10k \quad \text{for some integer } k. \]
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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.
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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: \[ \text{lcm}(10, 9) = 2^1 \times 3^2 \times 5^1 = 2 \times 9 \times 5. \]
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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.
- 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
\[ \boxed{90}. \]