List the first 20 positive solutions to each of the linear congruences in the sy

Question

List the first 20 positive solutions to each of the linear congruences in the system. In other words, you need to find the first 20 positive solutions of the congruence x 1(mod 2), then find the first 27 positive solutions of the congruence x 2(mod 3), and then the first 20 positive solutions of the congruence x 3(mod 5). Since a solution to the system is a number that satisfies all the congruences in the system, use your three lists to find two solutions to the system. Note: This is called the brute force approach.

Explanation / Answer

Here x=1(mod 2) means that difference (x-1) should be completely divided by 2.

So clearly if x=3 is one of its solution, because 3-1=2 that is completely divisible by 2.

Following the same pattern, 20 positive solutions of it are

3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41

And accordingly, x=2(mod 3) means that x must be a number so that x-2 should be divisible by 3, so if x= 5, then

5-2=3 and thus x= 2+ 3d where d is any integer.

So its 27 solutions will be , 5, 8, 11, 14, 17, 20,23,26,29,32,35,38,41,44,47,50,53,56,59,62,65,68,71,74,77,80,83

Accordingly third congruence given is set as x= 3+5d, so its different solutions will be given as :

8,13,18,23,28,33,38,43,48,53,58,63,68,73,78,83,88,93,98,103

Now we find that 23 is present in all the solutions. So clearly 23 is the required number that satisfies all the congruences in the system.

Answer

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