25) Which of the following numbers does NOT divide 100! (recall that 100! $=100 \cdot 99 \cdot$ $98 \cdots 3 \cdot 2 \cdot 1) ?$
(A) 1968
(B) 1988
(C) 1998
(D) 2008
(E) 2048
25) The answer is (D).

Let's factorize the 5 numbers:

$$
1968=2^{4} \cdot 3 \cdot 41 \quad 1988=2^{2} \cdot 7 \cdot 41 \quad 1998=2 \cdot 3^{3} \cdot 37 \quad 2008=2^{3} \cdot 251 \quad 2048=2^{11}
$$

from which:

$$
1968=3 \cdot 16 \cdot 41 \quad 1988=4 \cdot 7 \cdot 41 \quad 1998=2 \cdot 27 \cdot 37 \quad 2048=2 \cdot 4 \cdot 8 \cdot 32
$$

divide 100!, while 2008 cannot divide 100! because then the first 251 would divide it, but 251, being greater than 100, cannot be a factor of any number between 1 and 100.