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{\bf $\stackrel{\mbox{MA330}}{Sathaye}$ \hfill {\bf A few Ku\d{t}\d{t}aka Problems} \hfill \ \  }
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Here are some problems which provide practice for the solutions of the linear integer equations of the form $ax-by=c$.

\section{Class practice}
\begin{enumerate}
\item Find a number (and all such positive numbers) which is divisible by $29$ after adding $2$ to it and also
divisible by $45$ after subtracting $7$ from it.

Do determine the smallest positive such number.


\item Find a number divisible by $3$ after subtracting $5$ and divisible by $31$ after
subtracting $7$. ( A small variation from Bh\={a}skara I of 600 A.D.)

Do determine the smallest positive such number.

\item Find a number which has remainder $5$ when divided by $8$,
remainder $4$ when divided by $9$ and remainder $1$ when divided by $7$.

Do determine the smallest positive such number.

( A small variation from Bh\={a}skara I of 600 A.D.
 Bh\={a}skara I 600 A.D.)

\item Find a number which leaves a remainder $1$ when divided by $2,3,4,5$ 
or $6$ but is
divisible by $7$. 

Do determine the smallest positive such number.

(\={A}ryabha\d{t}\={i}ya bh\={a}\d{s}ya 600 A.D. ,
Ibn-al-Haitam 1000 A.D., Fibonacci 1202 A.D.)

\item Another variant is:
Find a number which leaves a remainder $1$ when divided by $2,3,4,5,6,7,8$ 
or $9$ but is
divisible by $11$. 
\item Suppose that if we count a set of objects by threes, then $2$ are left;
if we count by fives, then $3$ are left and if we count by sevens, 
then $2$ are left.
How many objects are there? 
(It is implied that you should find the smallest such positive number.
(Sunzi fifth century - the origin of the term  ``Chinese
Remainder Theorem!'')

\item The residue of the revolutions of Saturn is 24. Find the day number.
(laghubh\={a}skar\={i}ya  600 A.D.)
Explanation: Let $x$ be the day number or the number of days elapsed 
since the beginning
of the yuga  (actually the current Kali yuga). 

The planet Saturn makes  
$\frac{36,641}{394,479,375}$
revolutions in a day. . 
The statement that the ``residue of revolutions is
$24$'' means that the fractional number of revolutions is 
$\frac{24}{394,479,375}$.
So, your task is to find (smallest) number of days elapsed and 
corresponding number of
revolutions from the beginning of this yuga!

By the way, this number of days in a yuga, which formally has 
$432,000$ years gives
the average year as $365.259$ solar days in length.
The equation you get is 
$$36,641x-24 = 394,479,375y$$
 where $y$ is the number of complete revolutions.

\item The Sun's mean position was obtained as longitude $148$ deg. $20$ min.
Calculate the day number. (Mah\={a}bh\={a}skar\={i}ya 600 A.D.)
and the number of revolutions of the Sun.

For Sun, the reduced number of daily revolutions comes out to be 
$\frac{576}{210,389}$.
The fractional number of revolutions is calculated to be
$\frac{86,688}{210,389}$
after throwing away fractional part (less than 0.01) from the numerator.
Verify this from the longitude.

\end{enumerate}

\section{Some problems with reduction tricks}
\begin{enumerate}
\item Find a least positive integer $n$ which is divisible by $63$ and leaves a remainder of $90$ when divided by $100$.
(Bh\={a}skara II 1100+ AD).

\item Find a least positive integer $n$ such that $221n+65$ is divisible by $195$. 
(Bh\={a}skara II 1100+ AD).

\item Find a least positive integer $n$ such that $100n\pm 90$ is divisible by $63$. 
(Bh\={a}skara II 1100+ AD).

\item Find a least positive integer $n$ such that $5n-7$ is divisible by $63$ and $10n-14$ is also divisible by $63$.
 
(Bh\={a}skara II 1100+ AD).
\end{enumerate}


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{\bf Note:} It was customary to do astronomical calculations on a grand scale.
One use of these calculations is that by axiom, the planetary positions
were $0$ in the beginning and their current values can be established by
knowing their periods. Of course, everyone knows that corrections are always needed,
to both the formulas and observations.

Usually, the calculations start with the day number from the beginning of the year with
a given knowledge of the starting positions from record.

Here are some more amusing data.
The current yuga (Kali) has $432,000$ years and the previous three
(K\d{r}ta, Tret\={a} and Dv\={a}p\={a}ra ) were respectively $4,3,2$ times as long!
These four together are often called a (mah\={a})yuga. The current one has only $5104$ years
elapsed.

The creator Brahm\={a} was approximately $8$ and a half of {\it his} years old at the
beginning of the current yuga, which evaluates to $26782530123179$ years. There are other
variants of this huge number, though they all agree in the top significant digits.
According to another school of thought, he is however, about $50$ years old. 

The bigger divisions of time are:
A yuga (or mah\={a}yuga = the four together) is $4320000$ yrs.
A manu is $72$ such mah\={a}yugas. A kalpa is $14$ manus.
A day in the life of the creator is $2$ kalpas or $8709120000$ years!
It is amusing to compare this number with the number of years elapsed
since the big bang! The current theories seem to point to a number
between 11 and 20 billion years, so this number might match the
night and part of the day for Brahm\={a}!

During religious rituals, it is customary to recite the current year of the yuga,
the approximate age of Brahm\={a},
also the geographic position of the site, the current positions of various planets etc.
It is unfortunate that the same care was usually not taken by the authors of the books!


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