Question

when a number is devided by 13 the remainder is 11 and when the umber is devided by 17 the remainder is 9.find the number ?

Answer 1

128 is the number


13 x9 = 117 remains 11

17 × 7 = 119 remains 9

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Answer 2

Let x be the number.

Then,

x≡11mod13
x≡9mod17

Observe that 13,17 are co-prime integers. Then from Chinese remainder theorem we get,

[tex] x=((11*17*(17^{-1} mod13) + ( 9*13 * (13^{-1} mod17))) mod221 [/tex]

⟹x=(11×17×10)+(9×13×4) mod(221)

⟹x=(1870+468) mod221

⟹x=128
Therefore answer is 128.