Mathematics in Ancient and Medieval India c7.s5.122

c7.s5.122

From equation (1), we get x 5 ( mod 9 ) or x = 9u + 5 ...... (4), Subst. in equation (2), we get 8y₂ = 6(9u + 5) + 4 or 4y₂ = 27u + 17 From this u 1 (mod 4) or u = 4v + 1. Subst. this value of u in equation (4), we get x = 9 (4v + 1) + 5 or x = 36v + 14 . . . . . . (5) Subst. (5) in equation (3), we get, 7y₃ = 4x + 1 = 4(36v + 14) + 1 or 7y₃ = 144v + 57 . . . . . . (6) From here v - 2(mod 7) or v = 7p - 2. Hence from (3) and (6), 4x + 1 = 1008p - 231 or x = 252p - 58. Least value = 252 - 58 = 194.