2008 iTest Problems/Problem 25
Problem
A cube has edges of length cm. The cube gets chopped up into some number of smaller cubes, all of equal size, such that each edge of one of the smaller cubes has an integer length. One of those smaller cubes is then chopped up into some number of even smaller cubes, all of equal size. If the edge length of one of those even smaller cubes is
cm, where
is an integer, find the number of possible values of
.
Solution
The prime factorization of is
Note that
must be an integer, and since the cube is chopped up into smaller cubes of integral length,
must be one of the
factors of
. However, since the cube is divided twice, we have to divide 120 by two numbers greater than
so the numbers
and
would not work. Thus, there are
possible values of
.
See Also
2008 iTest (Problems) | ||
Preceded by: Problem 24 |
Followed by: Problem 26 | |
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