19) Anna and Marco have a collection of more than 40, but fewer than 80, postcards. Anna notices that the number of postcards minus 3 is a multiple of 8. Marco, on the other hand, notices that the number of postcards minus 1 is a multiple of 5. How many postcards do Anna and Marco have in total?
(A) Between 40 and 49
(B) between 50 and 59
(C) between 60 and 69
(D) between 70 and 79
(E) it is impossible: they must have miscounted.
19) The answer is (B). Let $n$ be the number of postcards that Anna and Marco have. We have $n=8a+3$ and $n=5b+1$ for suitable integers $a, b$. Since $40 \leq n \leq 79$, we must have $5 \leq a \leq 9$, that is, $n=43,51,59,67,75$. The only one of these numbers of the form $5b+1$ is 51.