To solve the problem of finding the smallest number that must be added to 18265 to make it a perfect square, follow these steps:
1. Identify the square root of the number:
- Calculate the square root of 18265. The square root of 18265 is approximately 135.16.
2. Find the next integer greater than the square root:
- The next integer greater than 135.16 is 136.
3. Calculate the square of this integer:
- The square of 136 is calculated as [tex]\( 136^2 = 18496 \)[/tex].
4. Determine the smallest addition required:
- Subtract the original number 18265 from 18496 to find the smallest number that must be added:
[tex]\[
18496 - 18265 = 231
\][/tex]
Thus, the smallest number that must be added to 18265 to make it a perfect square is [tex]\( 231 \)[/tex].
To verify:
- If we add 231 to 18265, we get: [tex]\( 18265 + 231 = 18496 \)[/tex].
- The square root of 18496 is indeed 136, which is an integer, thus confirming that 18496 is a perfect square.
Therefore, the smallest number that must be added to 18265 to make it a perfect square is 231.
