Final answer:
To make 18265 a perfect square, 96 must be subtracted, resulting in 18169, which is the square of 137. This is found using the long division method to find the largest perfect square less than 18265 and its square root.
Step-by-step explanation:
To find the least number that must be subtracted from 18265 to make it a perfect square, we need to determine the square root of 18265 using the long division method and identify the largest whole number that is a perfect square smaller than 18265.
Here is the step-by-step procedure:
- Starting with the number 18265, we pair the digits from right to left: (1) (82) (65).
- Find the largest square smaller than or equal to 1, which is 1 (1x1). Subtract this from the first pair, leaving us with 0. Bring down the next pair of digits (82), to make the remainder 082.
- The divisor now becomes 2 (1 doubled), with a blank space next to it. Choose the largest digit 'X' such that 2X*X is less than or equal to 082. We find that X=3 makes 2X*X = 63, which is less than 82. Place 3 above the pair (82) on the dividend line, and 63 below it. Subtract to get the new remainder 19.
- Bring down the next pair of digits (65) to form 1965.
- With the new divisor being 26 (13 doubled), we look for a digit X so that 26X*X is less than 1965. We find X=7 to fit, as 267*7 = 1869. Writing 7 at the top and 1869 at the bottom, we get a remainder of 96.
The square root of the largest perfect square less than 18265 is therefore 137, as 137*137 = 18769. To make 18265 a perfect square, we must subtract 96, because 18265 - 96 = 18169, which is the square of 137.
The least number that must be subtracted from 18265 to make it a perfect square is 96, and the square root of the resulting number, 18169, is 137.