In Eldorado a year has $20$ months, and each month has $20$ days. One day Brigi asked Adél who lives in Eldorado what day her birthday is. Adél answered that she is only going to tell her the product of the month and the day in her birthday. (For example, if she was born on the $19$th day of the $4$th month, she would say $4 \cdot 19 = 76$.) From this, Brigi was able to tell Adél’s birthday. Based on this information, how many days of the year can be Adél’s birthday?