What is the smallest number that must be subtracted from 63520 so that the resulting number becomes a perfect square?

Introduction / Context:
This problem focuses on perfect squares and asks how far a given integer is from the nearest perfect square below it. Such questions are important in number theory and help candidates gain intuition about square numbers and their spacing.

Given Data / Assumptions:
- We are given the number 63520.
- We must subtract the smallest possible non negative integer k so that 63520 − k is a perfect square.
- The resulting number must be exactly equal to n^2 for some integer n.

Concept / Approach:
To find the nearest perfect square less than or equal to a number N, we can compute the integer square root of N, that is the largest integer n such that n^2 ≤ N. Then we compare N with n^2. The required subtraction is N − n^2. If N is already a perfect square, this difference is zero.

Step-by-Step Solution:
Let N = 63520. Compute the integer square root of N. Find n such that n^2 ≤ 63520 and (n + 1)^2 > 63520. Compute 252^2 = 63504 and 253^2 = 64009. Since 63504 ≤ 63520 and 64009 > 63520, the closest perfect square below 63520 is 63504. The difference is 63520 − 63504 = 16, so we must subtract 16 to reach a perfect square.

Verification / Alternative Check:
Verify: 63520 − 16 = 63504, and we already know 252^2 = 63504. If we subtracted any smaller positive number, such as 14 or 24, the result would lie between 63504 and 63520 or above 63520 and would not coincide with a known perfect square. Therefore 16 is the minimal subtraction that works.

Why Other Options Are Wrong:
Subtracting 30 yields 63490, which is not a perfect square between 251^2 = 63001 and 252^2 = 63504.
Subtracting 24 gives 63496, again not equal to 251^2 or 252^2.
Subtracting 14 results in 63506, which lies between 252^2 and 253^2 but is not equal to any n^2.
Only subtracting 16 exactly hits 63504 which is 252^2.

Common Pitfalls:
Students often approximate square roots and may misidentify the nearest perfect square. Another mistake is to search both above and below the number when the question specifically asks for subtraction, implying the target square must be less than or equal to the original number. Careful computation of nearby squares is essential to avoid errors.

Final Answer:
The smallest number that must be subtracted from 63520 to make the result a perfect square is 16.