Quentin has 18 unit cubes. How many different rectangular prisms can he build if he uses all of the cubes.
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how many different rectangular prisms can you build if you use 18 cubes
To find out how many different rectangular prisms Quentin can build using the 18 unit cubes, we need to determine the number of distinct arrangements of those cubes.
One approach is to consider the prime factorization of 18, which is 2 * 3 * 3. This means that 18 can be expressed as the product of three distinct prime numbers, each with an exponent of 1.
Since the prime factorization of a number determines the number of distinct rectangular prisms, we can calculate this by finding all possible combinations of those prime factors.
For the prime factor 2, Quentin can choose to arrange the cubes in 2 different ways: 2 * 1 * 9 (2 long, 1 wide, and 9 high) or 2 * 3 * 3 (2 long, 3 wide, and 3 high).
For the prime factor 3, Quentin can rearrange the cubes in 2 ways: 3 * 1 * 6 (3 long, 1 wide, and 6 high) or 3 * 2 * 3 (3 long, 2 wide, and 3 high).
Finally, for the prime factor 3 again, Quentin has one option left, which is to arrange the cubes as 1 * 3 * 6 (1 long, 3 wide, and 6 high).
Therefore, Quentin can build a total of 6 different rectangular prisms using the 18 unit cubes.