113. Find the number which, when increased by 5 or decreased by 11, becomes a perfect square.

## Problems of Apastamba.

From the collection "Sulva-sutra“.

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113. According to the condition, we have: $N+5=u^{2}$ and $N-11=v^{2}$, from which $16=u^{2}-v^{2}=(u+v)(u-v)$, thus, we get two combinations:
I) $\left.\begin{array}{l}u+v=8 \\ u-v=2\end{array}\right\} u=5 ; v=3$ and $N=20$

II) $\left.\begin{array}{r}u+v=16 \\ u-v=1\end{array}\right\} u=\frac{17}{2} ; \quad v=\frac{15}{2}$ and $N=67 \frac{1}{4}$.

"Sulva-sutras" or "Rules of the Cord" - collections of rules concerning the construction of altars, their shapes, sizes, and orientations.

![](https://cdn.mathpix.com/cropped/2024_05_21_926f0bb262c8f9569516g-103.jpg?height=363&width=511&top_left_y=361&top_left_x=384)

Fig. 35. Currently, 3 such collections are known. Their authors are Baudhayana (6th or 7th century BCE), Katyayana, and Apastamba (4th or 5th century).