$35 \cdot 27$ If a 10 yuan note is exchanged for 1 jiao (0.1 yuan) coins and 2 jiao 5 fen (0.25 yuan) coins, then the total number of different ways to exchange the note, $n$, is
(A) 40.
(B) 38.
(C) 21.
(D) 20.
(E) 19.
(19th American High School Mathematics Examination, 1968)
[Solution] Let the number of 10 yuan notes converted into 1 jiao (0.1 yuan) coins and 2.5 jiao (0.25 yuan) coins be $p$ and $q$, respectively, then
$$
10 p + 25 q = 1000 \text{. }
$$

This can be rewritten as
$$
2 p = 5(40 - q) \text{. }
$$

The left side of the equation is a positive even number, so the right side $40 - q$ must also be even. Therefore, $q < 40$ must be an even number, and there are 19 such even numbers.
Thus, the answer is $(E)$.