What is the least number which is subtracted from 22541, the difference may be a perfect square ?

Question

Accepted Answer

To determine the least number that should be subtracted from 22541 so that the result is a perfect square, we need to find the largest perfect square less than or equal to 22541.

1. **Finding the largest perfect square less than 22541:**
   - First, estimate the square root of 22541. The square root of 22500 (a nearby perfect square) is 150, since $150^2 = 22500$.
   - Check $151^2$: $151 \times 151 = 22801$, which is greater than 22541.
   - Therefore, the largest perfect square less than 22541 is $150^2 = 22500$.

2. **Calculating the difference:**
   - Subtract 22500 from 22541: $22541 - 22500 = 41$.

3. **Conclusion:**
   - The least number that should be subtracted from 22541 to make it a perfect square is 41.

**Explanation for why other options are incorrect:**

- **Option 1 (17):** If 17 is subtracted from 22541, the result is 22524. The square root of 22524 is not an integer, so 22524 is not a perfect square.
  
- **Option 2 (21):** If 21 is subtracted from 22541, the result is 22520. The square root of 22520 is not an integer, so 22520 is not a perfect square.

- **Option 3 (31):** If 31 is subtracted from 22541, the result is 22510. The square root of 22510 is not an integer, so 22510 is not a perfect square.

Therefore, the correct answer is Option 4, as subtracting 41 from 22541 results in 22500, which is a perfect square.