+0

# Sum

+1
48
4

What is the remainder when 2 + 4 + 6 + ... + 100 + 102 + 104 is divided by 7?

Nov 2, 2021

#1
-5

The remainder is 1.

Nov 2, 2021
#2
+1
-3

this is just an arithmetic sequence

the sum of the entire sequence is the average of the first and last numbers of the sequence, times how many numbers there are.

(2 + 104)/2 = 53 times the numbers in the entire sequence ((104-2)+1 inclusive) =

53 times 103 = 5459

so when we divide, 5459 we get 779 with remainder:

6

Nov 2, 2021
#3
+3

∑[2 + 4 + 6 + ... + 100 + 102 + 104]==2,756 mod 7==5 -the remainder.

Nov 2, 2021
#4
+390
+3

Hello Guest,

use the double summation sign $$\displaystyle\sum$$ to get the answer ...

$$\displaystyle\sum_{n=1}^{52} 2n = 2756$$

$${\color{red}7},{\color{red}14},{\color{red}21},...,{\color{red}2751},{\color{blue}(2756)},{\color{red}2758}$$

So the remainder is $$(2756-2751=)$$ $${\color{blue}5}$$ .

Straight

Nov 6, 2021