A librarian receives 130 Mathematics books and 195 Portuguese books. She wants to arrange them on shelves, placing an equal number of books on each shelf, without mixing Mathematics and Portuguese books on the same shelf. How many books should she place on each shelf to minimize the number of shelves used?
Denoting by $n$ the number of books the librarian will place on each shelf, we have $130 \div n=$ the number of shelves for Mathematics books and $195 \div n=$ the number of shelves for Portuguese books. This shows that $n$ must be a common divisor of 130 and 195, since the number of shelves used is an integer. We know that when we increase the denominator of a fraction, the fraction decreases; for example, 27/10 is less than 27/8. Therefore, the larger the denominator $n$, the smaller the fractions $130 / n$ and 195/n will be, which means the fewer shelves will be used. Thus, $n$ must be the greatest common divisor (GCD) of 130 and 195. Since the prime factorizations of these numbers are $130=2 \times 5 \times 13$ and $195=3 \times 5 \times 13$, it follows that the GCD of 130 and 195 is $5 \times 13=65$.

Therefore, the librarian will place 65 books on each shelf, the number of shelves for Mathematics books is $130 \div 65=2$ and the number of shelves for Portuguese books is $195 \div 65=3$, which gives a total of $2+3=5$ shelves.