Find the smallest number to add to 1750 to make it a perfect square

Question

Find the smallest number that must be added to 1750 to make it a perfect square.

Solution — Step by Step

We need to find which two consecutive perfect squares 1750 falls between. Start by finding $\sqrt{1750}$ approximately.

We know $41^2 = 1681$ and $42^2 = 1764$ . Since $1681 < 1750 < 1764$ , the number 1750 lies between these two perfect squares.

The next perfect square after 1750 is $42^2 = 1764$ .

So we need to reach 1764 from 1750.

1764 - 1750 = 14

The smallest number we must add to 1750 to get a perfect square is 14.

1750 + 14 = 1764 = 42^2 \checkmark

The result 1764 is indeed a perfect square, and we cannot add any smaller positive integer to get a perfect square (since the previous perfect square 1681 is less than 1750).

Why This Works

Every number that is not a perfect square sits between two consecutive perfect squares. If $n^2 < x < (n+1)^2$ , we need to add $(n+1)^2 - x$ to reach the next perfect square.

We cannot add a smaller positive number because there is no perfect square between $n^2$ and $(n+1)^2$ — consecutive perfect squares have no perfect square between them.

Common Mistake

Many students confuse “smallest number to add” with “smallest number to subtract.” The question asks what to add, so we look at the next perfect square (42² = 1764), not the previous one (41² = 1681). If the question asked for the number to subtract, the answer would be $1750 - 1681 = 69$ .

Question

Solution — Step by Step

Why This Works

Common Mistake

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