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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