Time passed after New Moon Birth is called the age of the moon. Sighting is possible at different age in different months. So, age cannot be a criterion for sighting. Why is it so? Because, the orbit of the moon is elliptical and in its orbit, the moon moves faster when it is closer to the earth, and slower when it is farther from earth. When it moves faster, the moon becomes visible at smaller age (like 17 hours), and when it moves slower, it becomes visible at larger age (like 23 hours). The main factor that makes the moon visible is the angle between moon-earth-sun. When this angle becomes about 9 degrees, the moon starts to be visible. How much time it takes to get this angle depends upon the speed of the moon in its orbit.
(Courtesy of Moonsighting.com)