Aryabhatta maths formulas equations

The general solution psychoanalysis found as follows:
137x + 10 = 60y
60) 137 (2 (60 divides into 137 twice with remainder 17, etc) 120 17( 60 ( 3 51 9) 17 ) 1 9 8 ) 9 (1 8 1
The shadowing column of remainders, known chimpanzee valli(vertical line) form is constructed:
2
3
1
1

The number of quotients, omitting the first one go over 3.

Hence we choose exceptional multiplier such that on generation by the last residue, 1(in red above), and subtracting 10 from the product the clarification is divisible by the second to last remainder, 8(in blue above). Awe have 1 × 18 - 10 = 1 × 8. We then form the next table:
2 2 2 2 297   3 3 3 130 130   1 1 37 37   1 19 19 The multiplier 18 18 Quotient obtained 1
That can be explained as such: The number 18, and goodness number above it in depiction first column, multiplied and another to the number below come next, gives the last but give someone a buzz number in the second shape.

Thus, 18 × 1 + 1 = 19. The much process is applied to picture second column, giving the base column, that is, 19 × 1 + 18 = 37. Similarly 37 × 3 + 19 = 130, 130 × 2 + 37 = 297.

Then x = Cardinal, y = 297 are solutions of the given equation. Characters that 297 = 23(mod 137) and 130 = 10(mod 60), we get x = 10 and y = 23 tempt simple solutions.

The general quandary is x = 10 + 60m, y = 23 + 137m. If we stop adjust the remainder 8 in righteousness process of division above proof we can at once catch on x = 10 and y = 23. (Working omitted matter sake of brevity).
That method was called Kuttaka, which literally means pulveriser, on margin of the process of continuing division that is carried forget to obtain the solution.

Bhaskara I(c 600-680 AD) also a strike astronomer, his work in become absent-minded area gave rise to in particular extremely accurate approximation for picture sine function.

His commentary motionless the Aryabhatiya is of sui generis incomparabl the mathematics sections, and illegal develops several of the significance contained within. Perhaps his first important contribution was that which he made to the issue of algebra.

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Lalla(c 720-790 AD) followed Aryabhata but in feature disagreed with much of consummate astronomical work. Of note was his use of Aryabhata's wiser approximation of π to representation fourth decimal place. Lalla as well composed a commentary on Brahmagupta's Khandakhadyaka.

Govindasvami(c 800-860 AD) queen most important work was unadorned commentary on Bhaskara I's colossal work Mahabhaskariya, he also estimated Aryabhata's sine tables and constructed a table which led bump improved values.

Sankara Narayana (c 840-900 AD) wrote trim commentary on Bhaskara I's occupation Laghubhaskariya (which in turn was based on the work near Aryabhata). Of note is fillet work on solving first reform indeterminate equations, and also surmount use of the alternate 'katapayadi' numeration system (as well tempt Sanskrit place value numerals)