Find the least number that must be subtracted from 18265 so that the result is a perfect square.

Introduction / Context:
This question is about perfect squares and nearest square numbers. You are given a positive integer and asked to find how much must be subtracted from it so that the new number becomes a perfect square. Such problems check your familiarity with square numbers and your ability to approximate square roots and adjust to the nearest square below a given number.

Given Data / Assumptions:
- The given number is 18265.
- We want to subtract the smallest possible non negative integer from 18265 so that the result is a perfect square of some integer.
- We are concerned with the perfect square less than or equal to 18265, because subtracting moves us downward.

Concept / Approach:
The idea is to find the integer part of the square root of 18265. If n is the largest integer such that n^2 is less than or equal to 18265, then n^2 is the greatest perfect square not exceeding 18265. The difference 18265 - n^2 is the least number that must be subtracted to turn 18265 into a perfect square. Comparing it with the next higher square confirms that we have chosen the correct n.

Step-by-Step Solution:
Step 1: Estimate the square root of 18265. We know that 135^2 and 136^2 are close to this number.Step 2: Compute 135^2. The value is 135 * 135 = 18225.Step 3: Compute 136^2. The value is 136 * 136 = 18496.Step 4: Compare these squares with 18265. We see that 18225 is less than 18265, and 18496 is greater than 18265.Step 5: Since we must subtract to reach a perfect square, we focus on the largest perfect square less than 18265, which is 18225 = 135^2.Step 6: The least number that has to be subtracted is 18265 - 18225.Step 7: Compute the difference: 18265 - 18225 = 40.Step 8: Therefore, subtracting 40 from 18265 gives 18225, which is exactly 135^2 and thus a perfect square.

Verification / Alternative check:
If we subtract 40, we get 18225. Check that 18225 is a perfect square by taking its square root: 135 * 135 = 18225, so it is indeed a perfect square. If we tried subtracting any smaller number such as 38 or 30, the result would lie between 18225 and 18265 and therefore fall strictly between 135^2 and 136^2. No perfect square exists between these two consecutive squares, so no smaller subtraction would yield a perfect square.

Why Other Options Are Wrong:
If you subtract 30, you get 18235, which is not equal to 135^2 or 136^2. Similarly, subtracting 38 gives 18227, and subtracting 45 gives 18220. None of these results are perfect squares, because perfect squares around this region are 18225 and 18496 only. The option 47 also takes you to 18218, which again is not a square. Only subtracting 40 lands exactly on 18225, a perfect square.

Common Pitfalls:
Common mistakes include miscalculating the nearby squares or choosing the square above the given number and then trying to add instead of subtract. Some learners guess without checking the square values accurately, which can lead to off by one errors. Always confirm by squaring candidate integers around the estimated square root and verify which one gives the closest square below the given number.

Final Answer:
The least number that must be subtracted is 40.