What is the smallest integer that must be added to $1729$ so that the resulting number is a perfect square? State the integer and the perfect square obtained.

Step-by-step solution

1. Find the integer part of $\sqrt{1729}$.
2. Since $41^2=1681$ and $42^2=1764$, $\sqrt{1729}$ lies between $41$ and $42$.
3. The next perfect square greater than $1729$ is $42^2=1764$.
4. Calculate the difference: $1764-1729=35$.
5. Thus adding $35$ to $1729$ gives the perfect square $1764$, which equals $42^2$.

Concept tested: Completing to the next perfect square

Published 24 August 2026 · How this solution was prepared and checked

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