To find a number greater than 70 but less than 85 that is divisible by both 2 and 9, we first determine the least common multiple (LCM) of 2 and 9.
The LCM of 2 and 9 can be calculated as follows:
- The prime factorization of 2 is \(2^1\).
- The prime factorization of 9 is \(3^2\).
The LCM takes the highest power of each prime factor:
\[ \text{LCM}(2, 9) = 2^1 \times 3^2 = 2 \times 9 = 18 \]
Now, we need to find multiples of 18 that are greater than 70 and less than 85. We can calculate the multiples of 18:
- The first multiple of 18 is \(18 \times 1 = 18\).
- The second multiple is \(18 \times 2 = 36\).
- The third multiple is \(18 \times 3 = 54\).
- The fourth multiple is \(18 \times 4 = 72\).
- The fifth multiple is \(18 \times 5 = 90\).
Now, we check which of these multiples fall within the range of greater than 70 but less than 85:
- \(72\) is greater than \(70\) and less than \(85\).
- \(90\) is greater than \(85\), so it does not qualify.
Thus, the only number that fits the criteria is:
\[ \boxed{72} \]