20. Carl wrote a list of 10 distinct positive integers on a board. Each integer in the list, apart from the first, is a multiple of the previous integer. The last of the 10 integers is between 600 and 1000 . What is this last integer?
A 640
B 729
С 768
D 840
E 990
SolUtion
$\mathbf{C}$

The sequence of integers will have the form $q, q r, q r s, \ldots, q r s t u v w x y z$ with the first integer $q$ being successively multiplied by integers $r, s, \ldots, z$. The 10th integer in Carl's sequence, $F$, is then given by the product $F=$ qrstuvwxyz. Since the 10 integers are all distinct, none of $r, s, t, \ldots, z$ is 1 . Since $r, s, t, \ldots, z$ are integers, they are all at least 2 and hence $F \geq 2^{9}=512$. Therefore $q=1$ or we would have $F \geq 2^{10}=1024>1000$. All of $r, s, t, \ldots, z$ cannot be 2 since this would give $F=2^{9}=5121000$. Hence $q=1$, eight of the nine integers $r, s, t, \ldots, z$ are 2 and only one of them is greater than 2 . That integer must be 3 since otherwise $F \geq 1 \times 2^{8} \times 4=1024>1000$. Therefore the last integer in Carl's sequence is $1 \times 2^{8} \times 3=256 \times 3=768$