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What is the remainder when the sum $1 + 7 + 13 + 19 + \cdots + 253 + 259$ is divided by $6$?

 Jul 25, 2021
 #1
avatar+34388 
+2

Remainder = 2

 

The sum of the sequence is 5270      5270 / 6 = 878 R 2

 Jul 25, 2021
 #2
avatar+121055 
+2

1  +  7  +  13  +  19.........+  253    +  259

 

(1  + 259)   +  ( 7 + 253)  +  .......

 

We  have      ( 259 -1 )  / 6   + 1   =   44  terms

 

22 pairs of  them each sum to  260

 

So

 

22   *  260   =     5720

 

An even  number  whose  digits  sum to a multiple of  3  will be  divisible by  6

 

So

5718  digit sum  =  21   =   multiple of  3   thus divisible by 6

 

So

5720  /  6       leaves a remainder of 2

 

 

cool cool cool

 Jul 25, 2021
 #3
avatar+780 
+2

1+7+13+...+259

floor(259/6)=43

44(1)+43(6)=44+43*6=44(mod 6) = 2 (mod 6)

remainder is 2.

 Jul 25, 2021

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