Introduction:
To find the least number which must be added to 280564 so that the sum becomes a perfect square, we need to follow a systematic approach.

Method:
We can use the following steps to solve the problem:

Step 1: Find the square root of the given number.
The square root of 280564 is 530.

Step 2: Add 1 to the square root and square the result.
(530 + 1)^2 = 282249

Step 3: Find the difference between the result of Step 2 and the given number.
282249 - 280564 = 1685

Therefore, the least number which must be added to 280564 so that the sum becomes a perfect square is 1685.

Explanation:
The method used above is based on the fact that the difference between two consecutive perfect squares is always an odd number. We first find the square root of the given number, which gives us a rough estimate of the number required to make it a perfect square. Then, we add 1 to the square root and square the result to get the next perfect square. By subtracting the given number from this perfect square, we get the difference, which is the least number required to make the given number a perfect square.

Conclusion:
In conclusion, we can say that the least number which must be added to 280564 so that the sum becomes a perfect square is 1685.