To find a possible number of slices that the cake could have, we need to determine a number that is divisible by 3, 5, and 9. This means we need to find the least common multiple (LCM) of these numbers.
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Prime factorization of each number:
- \(3 = 3^1\)
- \(5 = 5^1\)
- \(9 = 3^2\)
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Determine the LCM:
- From the prime factorization, we take the highest power of each prime number:
- For the prime \(3\), the highest power is \(3^2\).
- For the prime \(5\), the highest power is \(5^1\).
Therefore, the LCM is calculated as follows: \[ \text{LCM} = 3^2 \times 5^1 = 9 \times 5 = 45 \]
- From the prime factorization, we take the highest power of each prime number:
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Conclusion: The smallest number of slices that the cake could have, which is divisible by 3, 5, and 9, is 45.
Thus, any multiple of 45 (like 90, 135, etc.) would also fit the conditions.
In summary, a possible number of slices that satisfies the condition is 45.