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Suppose that x is an integer that satisfies the following congruences:

3 + x == 2^2 (mod 3^3)

5 + x == 3^2 (mod 5^3)

7 + x == 4^2 (mod 7^3)

What is the remainder when x is divided by 105?

 Jan 30, 2022
 #1
avatar+117100 
+1

I've done this one before somewhere ... so has Heureka.

I'll start you off (although there may be an easier way)

 

3 + x == 2^2 (mod 3^3)

3 + x == 4 + 3^3K

x == 1 + 3^3K

this would also mean that

x == 1 + 3K

 

5 + x == 3^2 (mod 5^3)

5 + x == 9 +  5^5L 

 x == 4 +  5^3L 

x == 4+ 5L

x == -1 + 5L

 

7 + x == 4^2 (mod 7^3)

7 + x == 16 + 7^3M

x == 9 + 7M

x == 2 +7M

 

First I solved  1+3K = -1+5L      and got   x= -1 mod 15

 

Then I solved    x=1+15a  with  x = 2+7M  and got   x=79+105b

 

So I got an answer of 79 but I could easily have made a careless error.

 

I use the Euclidean Algorithm to solve the diophantine equations.  

 Jan 31, 2022

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