SOLUTION: Prove or disprove: There exists an integer a for which 20a ≡ 2 mod 8.

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Question 1178231: Prove or disprove: There exists an integer a for which 20a ≡ 2 mod 8.
Answer by ikleyn(52776) About Me  (Show Source):
You can put this solution on YOUR website!
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Let "a" be such an integer number that  20a ≡ 2 (mod 8)


It means that  20a - 2 is divisible by 8:  


      20a-2 = 8m  for some integer m      (1)


Divide by 2 both sides of equation  (1).  You will get then


    10a - 1 = 4m.


But it is just a contradiction:  10a is a multiple of 2;  4m  is a multiple of 2;

hence, their difference 1 = 10a - 4m  must be multiple of 2, but it is not the case.


The contradiction proves that 20a ≡ 2 mod 8 is not possible.