How many integers from 1 to 1000 (exclusive) are divisible by the cube of an integer larger than or equal to 2?

To find the number of integers from 1 to 1000 (exclusive) that are divisible by the cube of an integer larger than or equal to 2, we can follow these steps:

Step 1: Find the cubes of integers larger than or equal to 2.
- Start with the number 2 and cube it (2^3 = 8).
- Proceed to the next integer, 3, and cube it (3^3 = 27).
- Continue this process until we reach an integer whose cube exceeds 1000.

Step 2: Count the number of integers in the range that are divisible by each cube.
- For each cube, count how many integers from 1 to 1000 (exclusive) are divisible by that cube.
- To do this, divide 1000 by the cube and round down to the nearest whole number.

Step 3: Sum up the counts from step 2 to get the final answer.

Let's go through these steps to find the solution.

Step 1:
The cubes of integers larger than or equal to 2 are:
2^3 = 8
3^3 = 27
4^3 = 64
5^3 = 125
6^3 = 216
7^3 = 343
8^3 = 512
9^3 = 729
10^3 = 1000

Now we have a list of cubes up to the largest cube that is less than or equal to 1000, which is 729.

Step 2:
Now we will count the number of integers from 1 to 1000 (exclusive) that are divisible by each cube:

- For 8 (2^3), we can divide 1000 by 8 and round down to the nearest whole number, which is 125.
- For 27 (3^3), we can divide 1000 by 27 and round down to the nearest whole number, which is 37.
- For 64 (4^3), we can divide 1000 by 64 and round down to the nearest whole number, which is 15.
- For 125 (5^3), we can divide 1000 by 125 and round down to the nearest whole number, which is 8.
- For 216 (6^3), we can divide 1000 by 216 and round down to the nearest whole number, which is 4.
- For 343 (7^3), we can divide 1000 by 343 and round down to the nearest whole number, which is 2.
- For 512 (8^3), we can divide 1000 by 512 and round down to the nearest whole number, which is 1.
- For 729 (9^3), we can divide 1000 by 729 and round down to the nearest whole number, which is 1.

Step 3:
To find the total number of integers from 1 to 1000 (exclusive) that are divisible by the cube of an integer larger than or equal to 2, sum up the counts from step 2:
125 + 37 + 15 + 8 + 4 + 2 + 1 + 1 = 193.

Therefore, there are 193 integers from 1 to 1000 (exclusive) that are divisible by the cube of an integer larger than or equal to 2.