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Find x for the CRT when x= 2 mod 3; x= 3 mod 5; x = 2 mod 7

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Find x for the CRT when x= 2 mod 3; x= 3 mod 5; x = 2 mod 7

(a) 33

(b) 22

(c) 23

(d) 31

I got this question by my school principal while I was bunking the class.

I would like to ask this question from Number Theory in portion More Number Theory of Cryptograph & Network Security

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The correct choice is (c) 23

For explanation: We have M = 3 x 5 x 7 = 105; M/3 = 35; M/5 = 21; M/7 = 15.

The set of linear congruences

35 x b1 = 1 (mod 3); 21 x b2 = 1 (mod 5); 15 x b3 = 1 (mod 7)

has the solutions b1 = 2; b2 = 1; b3 = 1. Then,

x = 2 x 2 c 35 + 3 x 1 x 21 + 2 x 1 x 15 = 233 (mod 105) = 23.

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