An event was held in the month of June but James forgot on which date. To refresh his memory, he asked five of his colleagues about what was the precise date. The responses were as follows:

Abe: The date was an odd number.
Belinda: It was greater than 13.
Charlene: It was not a perfect square.
Daphne: It was a perfect cube.
Ethan: It was less than 17.

The next day James was able to get a copy of the yearly schedule, whereby he noticed that exactly one of the five statements was true.

What was the date of the event?