In this number theory question, you must count how many numbers from 2000 to 7000 are both perfect squares and perfect cubes at the same time.

Introduction / Context:
This problem tests understanding of perfect squares and perfect cubes, and how to recognize numbers that are both. Such numbers are called perfect sixth powers because a number that is simultaneously a square and a cube must be a power of an integer raised to the sixth power. The question asks you to identify how many such numbers lie between 2000 and 7000, inclusive of the endpoints if they qualify.

Given Data / Assumptions:

• The range of interest is from 2000 to 7000.
• We are looking for numbers that are both perfect squares and perfect cubes.
• There are no additional restrictions such as odd, even, or prime.
• All numbers are understood to be positive integers.

Concept / Approach:
A number that is both a perfect square and a perfect cube must be of the form n^6, where n is a positive integer. This is because being a perfect square means the exponent on the prime factorization is even, being a cube means it is a multiple of 3, and the least common multiple of 2 and 3 is 6. So the central idea is to find all integers n such that n^6 lies between 2000 and 7000. Then we count how many such n exist.

Step-by-Step Solution:
Step 1: Express the condition as 2000 ≤ n^6 ≤ 7000. Step 2: Estimate powers of small integers. Compute 2^6 = 64, which is far below 2000. Step 3: Compute 3^6 = 729, which is still below 2000, so not in the range. Step 4: Compute 4^6. First 4^3 = 64, then square that to get 64^2 = 4096, which lies between 2000 and 7000. Step 5: Compute 5^6. First 5^3 = 125, then square that to get 125^2 = 15625, which is greater than 7000. Step 6: For n bigger than 5, n^6 will be even larger, so no further values will fall inside the range. Step 7: Check n = 4 carefully: 4096 is indeed both a perfect square (64^2) and a perfect cube (16^3), so it is valid. Step 8: Conclude that only one number, 4096, satisfies all the conditions.

Verification / Alternative check:
You can double check with the perfect square and cube perspective. 4096 is 64^2 so it is a square. It is also 16^3 so it is a cube. There is no other perfect cube between 2000 and 7000 that is also a perfect square. For instance, 10^3 = 1000 and 20^3 = 8000, so only 11^3 to 18^3 lie in the numeric range, and none of those are perfect squares. This reinforces that 4096 is the unique candidate.

Why Other Options Are Wrong:
Option 0 would suggest there is no such number, but 4096 contradicts that. Options 2, 3, and 4 would require at least two or more distinct sixth powers inside the interval, but we have seen that moving from 4^6 to 5^6 already passes beyond 7000. Since only 4^6 lies in the given range, any count higher than 1 is impossible.

Common Pitfalls:
Students sometimes confuse perfect squares and perfect cubes and look for any square or any cube rather than numbers that are both. Another common mistake is to test random numbers between 2000 and 7000 instead of using the sixth power idea, which wastes time and can lead to errors. Some learners also compute powers incorrectly, especially 4^6 and 5^6, so careful calculation is important. Recognizing that being both a square and a cube immediately implies a sixth power makes this question much easier and more systematic.

Final Answer:
There is exactly 1 number between 2000 and 7000 which is both a perfect square and a perfect cube.