Emma is using connecting cubes to make a rectangular model with a height of one connecting cube. With her cubes she can make the model in exactly 3 different ways. If Emma has more than 12 but fewer than 17 cubes, how many cubes does she have?
16 cubes
16
Emma is using connecting cubes to make a rectangular model with a height of one connecting cube. With her cubes, she can make the model in exactly 3 different ways. If Emma has more than 12 but fewer than 17 cubes, how many cubes does she have?
To solve this problem, we need to find a number between 12 and 17 that can be evenly divided into 3 different ways to make a rectangular model with a height of one connecting cube.
Let's start with the minimum number of cubes, which is 13. We can divide 13 cubes into a rectangular model in the following ways:
1 x 1 x 13
1 x 13 x 1
13 x 1 x 1
So, 13 is a possible solution.
Now let's move on to the next number, which is 14. We can divide 14 cubes into a rectangular model in the following ways:
1 x 1 x 14
1 x 14 x 1
2 x 7 x 1
Only two different ways, so 14 is not a solution.
Next, we check 15 cubes:
1 x 1 x 15
1 x 15 x 1
3 x 5 x 1
Again, only two different ways, so 15 is not a solution.
Now let's move on to 16 cubes:
1 x 1 x 16
1 x 16 x 1
2 x 8 x 1
4 x 4 x 1
Four different ways, so 16 is a possible solution.
Finally, we check 17 cubes:
1 x 1 x 17
1 x 17 x 1
Only two different ways, so 17 is not a solution.
From our calculations, we can see that Emma has either 13 or 16 cubes since these are the only numbers that can be divided into 3 different ways to make a rectangular model with a height of one connecting cube.
Therefore, Emma has either 13 or 16 cubes.