Question

Find the smallest number that needs to be subtracted by 180 to get a perfect square.

Option 1: 25
Option 2: 35
Option 3: 45
Option 4: 55

Answer 1

The smallest number that needs to be subtracted by 180 to obtain a perfect square is Option 2: 35.

Step-by-step explanation:

To find the smallest number that needs to be subtracted by 180 to get a perfect square, let's denote the unknown number as
`\( x \)`. The given condition can be expressed as the equation
`\( x - 180 = n^2 \)` , where
`\( n \)` is a positive integer representing the square root.

Solving for
`\( x \),` we get
`\( x = n^2 + 180 \)` . We need to minimize
`\( n \)` to find the smallest
`\( x \)`. Notice that
`\( n^2 \)` will always be a perfect square, so the key is to minimize the value of
`\( n \)`.

Consider the given options:

- Option 1:
`\( 25^2 + 180 = 625 + 180 = 805 \)`

- Option 2:
`\( 35^2 + 180 = 1225 + 180 = 1405 \)`

- Option 3:
`\( 45^2 + 180 = 2025 + 180 = 2205 \)`

- Option 4:
`\( 55^2 + 180 = 3025 + 180 = 3205 \)`

Among these, Option 2 yields the smallest result,
`\( x = 1405 \)` , making it the correct answer.

In summary, the smallest number that needs to be subtracted by 180 to get a perfect square is obtained by choosing the option with the minimum
`\( n \),` and in this case, it is Option 2.