How many integers from 1 to 1000 (exclusive) are divisible by the cube of an integer larger than or equal to 2?
such cubes:
8,27,64,125,216,343,512,729
So, figure out how many multiples of these numbers there are, removing duplicates.
i calculated 191.
but the answer is something else.
please help
Whats the answe?
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 use the concept of divisibility.
Let's break it down into steps:
1. Determine the largest cube that is less than or equal to 1000. The largest cube less than or equal to 1000 is 8^3 = 512, because 9^3 = 729 is larger than 1000.
2. Create a list of numbers that form the cube of an integer larger than or equal to 2. In this case, we have (2^3, 3^3, 4^3, 5^3, 6^3, 7^3, 8^3) = (8, 27, 64, 125, 216, 343, 512).
3. Count the numbers divisible by each cube in the list. We start by considering 8. We count the numbers from 1 to 1000 that are divisible by 8, which is 1000/8 = 125. Then, we count the numbers that are divisible by 27, 64, 125, 216, 343, and 512 individually.
4. Sum up the counts from step 3 to get the final answer, which will be 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.
Therefore, to solve this problem, you need to find the counts individually for each cube (8, 27, 64, 125, 216, 343, and 512), then add them up to get the final answer.