Professor Newton divided his students into groups of 4 and 2 were left over. He divided his students into groups of 5 and one student was left out. If 15 students are women and there are more women than men, the number of male students is:
(a) 7 ;
(b) 8 ;
(c) 9 ;
(d) 10 ;
(e) 11 .
The correct option is (c).

Since the number of male students is less than 15 and the number of female students is 15, we have $15<$ male students + female students $<15+15=30$, which means the total number of students is between 15 and 30.

Solution 1: When we divide the number of students by 4, there are 2 students left, so the number of students is even. When we divide by 5, there is a remainder of 1, so the last digit of the number of students is 1 or 6; being even, it can only be 6. Thus, we have only two possible values, 16 or 26. We discard 16 because it is divisible by 4, so the number of students is 26. Consequently, we have $26-15=9$ male students.

Solution 2: Observe that the number 6, when divided by 4, leaves a remainder of $2 \text{ and }$, when divided by 5, leaves a remainder of 1. Therefore, if we add 6 to a common multiple of 4 and 5, the resulting number will also have this property. The least common multiple of 4 and 5 is 20, so the possible values for the number of students are $6, 26, 46, 66, \ldots$ Since the number of students is between 15 and 30, this number is 26, and it follows that we have $26-15=9$ male students.