Use Chinese Remainder Theorem to find the value of x such that x = 0 ( MOD 2) , x = 2 (MOD 3) and x = 3 (MOD 5).

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Kulchandra Gautam · Accepted Answer

Given : a1 = 0 m1 = 2 a2 = 2 m2 = 3 a3 = 3 m3 = 5 To find the value of x : we will use the solution, x = a1 M1 Y1 + a2 M2 Y2 + a3 M3 Y3 _ _ _ _ _ ak Mk Yk ( MOD M) M = m1 x m2 x m3 = 2 x 3 x 5 = 30 M1 = M / m1 M2 = M / m2 M3 = M / m3 = 30 / 2 = 30 / 3 = 30 / 5 = 15 = 10 = 6 now, finding the value of Y1 , Y2 and Y3 M1 Y1 = 1 MOD m1 M2 Y2 = 1 MOD m2 M3 Y3 = 1 MOD m3 15 Y1 = 1 MOD 2 10 Y2 = 1 MOD 3 6 Y3 = 1 MOD 5 Y1 = 1 Y2 = 1 Y3 = 1 now , putting the values we get x = a1 M1 Y1 + a2 M2 Y2 + a3 M3 Y3 ( MOD M ) = 0 x 15 x 1 + 2 x 10 x 1 + 3 x 6 x 1 ( MOD 30 ) = 0 + 20 + 18 ( Mod 30 ) = 38 ( Mod 30 ) now putting the value of x , To verify: i) x = 0 ( MOD 2 ) ii) x = 2 ( MOD 3 ) ii) x = 3 ( MOD 5 ) 2 | 38 | 19 3 | 38 | 12 5 | 38 | 7 - 38 - 36 - 35 ------------- ------------ ---------- 0 ( True ) 2 ( True ) 3 ( True )

Use Chinese Remainder Theorem to find the value of x such that x = 0 ( MOD 2) , x = 2 (MOD 3) and x = 3 (MOD 5).

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Use Chinese Remainder Theorem to find the value of x such that x = 0 ( MOD 2) , x = 2 (MOD 3) and x = 3 (MOD 5).

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