176 7. ANCIENT NUMBER THEORYsolutions of the equation Ax = 23?/+ 5. First, we carry out the Euclidean algorithm
until 1 appears as a remainder:We then write the quotients (5 and 1 in this case) from the Euclidean algorithm
in a column, and beneath them we write the additive term c if the number of
quotients is even (in this case, two), otherwise -c. At the bottom of the column
we write a 0. This zero is inserted so that the same transformation rule applies at
the beginning as in all other steps of the algorithm. We then reduce the number
of rows successively by operating on the bottom three rows at each stage. The
second-from-last row is replaced by its product with the next-to-last row plus the
last row; the next-to-last row is simply copied, and the last row is discarded. Thus
to solve this system the kuttaka method amounts to the transformations
This column now gives χ and y, and indeed, 4-30 = 23-5+5. Diophantus showed how
to find a particular solution of such a congruence. Brahmagupta, however, found
all the solutions. He took the solutions χ and y obtained by the kuttaka method,
which were generally quite large numbers, divided χ by b and y by a, replaced them
by the remainders, and gave the general χ and y as a pair of arithmetic sequences
with differences b and a, respectively. In the present case, the general solution is
χ = 30 + 23fc, y = 5 + 4k. The smallest positive solution χ = 7, y = 1, is obtained
by taking k = — 1.
Brahmagupta's rule for finding the solutions is more complicated than the
discussion just given, since he does not assume that the numbers á and b are
relatively prime. However, the greater generality is only apparent. If the greatest
common divisor of á and 6 is not a factor of c, the problem has no solution; if it is
a factor of c, it can be divided out of the problem.
Brahmagupta also considers such equations with negative data and is not in
the least troubled by this complication. It seems clear that the name pulverizer
was applied because the original data are repeatedly broken down by the Euclidean
algorithm (they are "pulverized").
Astronomical applications. It was mentioned above that this kind of remainder
arithmetic, which we now call the theory of linear congruences, has applications to
the calendar. Brahmagupta proposed the problem of finding the (integer) number
of elapsed days when Jupiter is 22°, 30' into the sign of Aries^10 (Colebrooke, 1817,
p. 334). Brahmagupta converted the 22°, 30' into 1350'. He had earlier taken the
sidereal period of Saturn to be 30 years and to be a common multiple of all cycles.
(^10) Obviously, Jupiter will pass this point once in each revolution, but it will reach exactly this
point at the expiration of an exact number of days (no fractional hours or minutes) only once in
a yuga, which is a common period for all the heavenly bodies. Brahmagupta took 30 years as a
yuga, but his method is general and will yield better results if a more accurate yuga is provided by
observation. He says that the value of 30 years is given for a yuga only to make the computation
easier.
23 13 = 5-4 + 3,
4 = 1 -3+1.
5
1
5
0