Thank you in advance! (Chinese Remainder Theorem) Given n congruences x ? a1( mo

Question

Thank you in advance!

(Chinese Remainder Theorem) Given n congruences x ? a1( mod m1), ..., x ? an( mod mn), where gcd mimj = 1 for each i ?= j. The given algorithm finds a unique solution x such that 0 ? x < m where m = m1...mn.

(a) For each i find bi such that (m/mi)bi ? 1( mod mi). 1

(b) Set x = (m/m1)b1a1 + ... + (m/mn)bnan.

(c) If x is not in the proper range, then add or subtract a multiple of m.

Find a unique solution to the given set of congruence. x ? 34( mod 9). x ? 23( mod 10) (10 points).

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Explanation / Answer

Answer:

Given set of congruence is,

From the above given congruence’s from the Chinese remainder theorem, let

According to the given rules, need to find the gcd of 9 and 10.

So, the gcd of 9 and 10 is given as,

Therefore,

Now, let and chose some such that

Substitute

Substitute

So,

Therefore, the unique solution is .

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