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Find the remainder when \(3 \times 13 \times 23 \times 33 \times \ldots \times 183 \times 193\) is divided by 5.

tertre  Mar 22, 2017
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 #1
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(3×13×23×33×...×183×193) = product_(n=1)^20(-7 + 10 n) = 9936127455089061347552058319626135201 mod 5 =1

Guest Mar 22, 2017
 #2
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+1

(0 + 3) (10 + 3) =

(30 + 9)(20 + 3)=

(600 + 180 + 90 + 27) (30 + 3) =

(..............+ 81)(40 + 3).......   etc.

 

So  (3 x 13 x 23 x........183 x 193) / 5   will have a remainder =

 

3^20 mod 5   =  

 

(3^4)^5 mod  5   =

 

(3^4) mod 5  =  

 

1

 

 

cool cool cool

CPhill  Mar 22, 2017

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