AoPS Academy
, in square units, is![\[ \textbf{(A)}\ 24 \qquad
\textbf{(B)}\ 30 \qquad
\textbf{(C)}\ 46 \qquad
\textbf{(D)}\ 66 \qquad
\textbf{(E)}\ 74
\]](http://latex.artofproblemsolving.com/8/2/f/82ffc15072ca99f6cd1abc312781eb4827817356.png)
[hide=P#]a1b2c3[/hide] [hide=S#]a1b2c3[/hide]
is rotated 
degrees about one side. The space through which the hexagon travels during the rotation forms a solid. Find the volume of this solid. is rotated degrees about one side. The space through which the octagon travels through during the rotation forms a solid. Find the volume of this solid.
grid is made. 
unit squares are created by separating the side with length 
in portions and separating the side with length 
in portions. Find, with proof, a formula for how many rectangles with sides parallel to the grid can be created in terms of and . For example, when 
and 
, it can be found by casework that 
rectangles can be created.
:o
so 
and since 
is just 
therefore 
so 
and height 
:
Volume of cone frustum with base radii 
,
have the digit 
and are divisible by 
?
now, we have covered all the numbers in the 7 ones place. (27, if I counted right) now, lets find numbers with tens digit 7 that do NOT have 7 in its one place, since we already counted the _77 in the first case.
this totals up to 25 numbers, so the total is
so 
?
without a calculator.
for 
:
:
.
and 
where the rest has been omitted because of the floor function. Now 
and multiplying by 
yields 
grid, each unit square is created by dividing the sides of lengths 
and 
.
horizontal lines and 
vertical lines:
that can be formed, we multiply the number of ways to choose the horizontal lines and the ways to choose the vertical lines:
satisfy 
,
and 
are positive integers, what is the probability that 
is divisible by 
?
,
have?
not 
has the most factors?
?
.
,
represents a distinct digit from 
to 
.
is our answer. (By python, 
does not work and surprisingly, 
is the only permutation number that is divisible by 
8907