Question 1208759
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The goal is to solve this system of congruences
x = 5 (mod 45)
x = 2 (mod 72)
x = 9 (mod 999)


Recall that if a = b (mod n) then a-b = nk for some integer k.
Rearranging things gives a = nk + b.
Use this idea to transform the first two equations into x = 45k+5 and x = 72m+2. We cannot re-use k.


Both equations involve x, so equate the right hand sides and we get,
72m+2 = 45k+5
72m-45k = 5-2
9(8m-5k) = 3
9(integer-integer) = 3
9*(integer) = 3
integer = 3/9
integer = 1/3
which is a contradiction. The value 1/3 = 0.33333... is not in the set of integers.


Therefore the first two equations of the original system do not have a solution.
Overall the entire system doesn't have a solution either.


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I used a Python script to check integers from x = 1 to x = 10,000,000 and couldn't find any solutions. 


This numeric approach of course doesn't prove there aren't any solutions, since there are infinitely many integers to check, but it's useful to get partial backup confirmation.


A spreadsheet is another alternative verification route.
You may be asking yourself "Can we use Chinese Remainder Theorem?"
The answer would be "No because the mod values 45, 72, and 999 are not pairwise coprime."


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Answer: <font color=red>No solutions</font>
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