Finding the Largest Four-Digit Perfect Cube

The problem asks us to find the largest number that has exactly four digits and is also a perfect cube.

A perfect cube is a number that can be obtained by multiplying an integer by itself three times (i.e., raising an integer to the power of 3).

Four-digit numbers are integers ranging from 1000 to 9999, inclusive.

To find the largest four-digit perfect cube, we need to find the largest integer whose cube is less than or equal to 9999.

Let's start cubing integers and see when the result exceeds 9999.

• We know that $10^3 = 1000$. This is a four-digit number and a perfect cube.
• Let's try larger integers. We can estimate by considering that $20^3 = 8000$, which is a four-digit perfect cube.
• Let's try the next integer, 21:
• $21^3 = 21 \times 21 \times 21$
• First, calculate $21 \times 21$: $21 \times 21 = 441$
• Now, multiply 441 by 21: $441 \times 21$

So, $21^3 = 9261$. This is a four-digit number (since $1000 \le 9261 \le 9999$) and it is a perfect cube.

Now, let's check the next integer, 22:

• $22^3 = 22 \times 22 \times 22$
• First, calculate $22 \times 22$: $22 \times 22 = 484$
• Now, multiply 484 by 22: $484 \times 22$

So, $22^3 = 10648$. This is a five-digit number (since $10648 > 9999$).

This means that any integer greater than or equal to 22, when cubed, will result in a number with five or more digits. Therefore, the largest integer whose cube is a four-digit number is 21.

The largest four-digit perfect cube is $21^3 = 9261$.

Let's compare this with the given options:

• Option 1: 9261. This is a perfect cube ($21^3$) and is a four-digit number.
• Option 3: 8000. This is a perfect cube ($20^3$) and is a four-digit number, but it is smaller than 9261.
• Option 4: 9999. This is a four-digit number, but as we found, the largest perfect cube is 9261, and the next is 10648, so 9999 is not a perfect cube.

Thus, 9261 is indeed the largest four-digit number which is a perfect cube.