7) How many perfect squares divide 1600? [A perfect square is a number of the type $n^{2}$, where $n$ is a natural number. $1,4,9,16$, are examples of perfect squares.
(A) 2 ,
(B) 4 ,
(C) 8 ,
(D) 10 ,
(E) 12 .
7. The answer is $(\mathbf{C})$.

If we factorize 1600 into prime factors, we find

$$
1600=2^{6} \cdot 5^{2} .
$$

Therefore, the divisors of 1600 are numbers that can be written in the form

$$
2^{p} \cdot 5^{q},
$$

where $p$ is an integer between 0 and 6, inclusive, and $q$ is an integer between 0 and 2, inclusive. Among these, the perfect squares are those for which: (1) both $p$ and $q$ are even; (2) one of them is even and the other is zero; (3) both are zero. Therefore, the perfect squares that divide 1600 are:

$$
\begin{array}{r}
2^{0} \cdot 5^{0}=1,2^{2} \cdot 5^{0}=4,2^{4} \cdot 5^{0}=16,2^{6} \cdot 5^{0}=64 \\
2^{0} \cdot 5^{2}=25,2^{2} \cdot 5^{2}=100,2^{4} \cdot 5^{2}=400,2^{6} \cdot 5^{2}=1600
\end{array}
$$

which are 8 in total.

[Problem proposed by X.Y. Lu.]