The two-digit numbers that are less than 80 and greater than 65 are 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, and 79. Among these, we need to find a number that is divisible by both 2 and 9.
A number is divisible by 2 if it is even, and it is divisible by 9 if the sum of its digits is a multiple of 9.
Let's check the even numbers in the range:
- 66: \( 6 + 6 = 12 \) (not divisible by 9)
- 68: \( 6 + 8 = 14 \) (not divisible by 9)
- 70: \( 7 + 0 = 7 \) (not divisible by 9)
- 72: \( 7 + 2 = 9 \) (divisible by 9)
Since 72 is divisible by both 2 and 9, the required number is 72.