To solve the problem step by step, we need to analyze the conditions given in the question:
1. **Identify the characteristics of the number**:
- It is a two-digit even number.
- It is a common multiple of 3, 4, and 6.
- It has a total of 9 factors.
2. **List the two-digit even numbers**:
- The two-digit even numbers range from 10 to 98. They are: 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98.
3. **Find common multiples of 3, 4, and 6**:
- The least common multiple (LCM) of 3, 4, and 6 can be calculated:
- The prime factorization of 3 is \(3^1\).
- The prime factorization of 4 is \(2^2\).
- The prime factorization of 6 is \(2^1 \times 3^1\).
- The LCM is obtained by taking the highest power of each prime:
- \(2^2\) from 4 and \(3^1\) from 3 or 6.
- Thus, LCM = \(2^2 \times 3^1 = 4 \times 3 = 12\).
- Therefore, the common multiples of 3, 4, and 6 are multiples of 12. The two-digit multiples of 12 are: 12, 24, 36, 48, 60, 72, 84, 96.
4. **Identify which of these numbers have 9 factors**:
- To find the number of factors of a number, we use its prime factorization. The formula for the number of factors is:
\[
(e_1 + 1)(e_2 + 1) \ldots (e_n + 1)
\]
where \(e_i\) are the powers of the prime factors.
- Now, we will check the numbers: 12, 24, 36, 48, 60, 72, 84, 96.
- **For 12**:
- Prime factorization: \(2^2 \times 3^1\)
- Number of factors: \((2 + 1)(1 + 1) = 3 \times 2 = 6\)
- **For 24**:
- Prime factorization: \(2^3 \times 3^1\)
- Number of factors: \((3 + 1)(1 + 1) = 4 \times 2 = 8\)
- **For 36**:
- Prime factorization: \(2^2 \times 3^2\)
- Number of factors: \((2 + 1)(2 + 1) = 3 \times 3 = 9\)
- **For 48**:
- Prime factorization: \(2^4 \times 3^1\)
- Number of factors: \((4 + 1)(1 + 1) = 5 \times 2 = 10\)
- **For 60**:
- Prime factorization: \(2^2 \times 3^1 \times 5^1\)
- Number of factors: \((2 + 1)(1 + 1)(1 + 1) = 3 \times 2 \times 2 = 12\)
- **For 72**:
- Prime factorization: \(2^3 \times 3^2\)
- Number of factors: \((3 + 1)(2 + 1) = 4 \times 3 = 12\)
- **For 84**:
- Prime factorization: \(2^2 \times 3^1 \times 7^1\)
- Number of factors: \((2 + 1)(1 + 1)(1 + 1) = 3 \times 2 \times 2 = 12\)
- **For 96**:
- Prime factorization: \(2^5 \times 3^1\)
- Number of factors: \((5 + 1)(1 + 1) = 6 \times 2 = 12\)
5. **Conclusion**:
- The only number that meets all the criteria (two-digit even number, common multiple of 3, 4, and 6, and has exactly 9 factors) is **36**.
### Final Answer:
The number is **36**.
