\(1+5+5^2+...+5^{404}\)
\(=5^3\left(1+5+5^2\right)+5^4\left(1+5+5^2\right)+...+5^{404}\left(1+5+5^2\right)\)
\(=\left(1+5+5^2\right)\left(5^3+5^4+...+5^{403}+5^{404}\right)\)
\(=31.\left(5^3+5^4+...+5^{403}+5^{404}\right)\)
Vậy tổng trên chia hết cho 31
1 + 5 + 52 + .... + 5404
= ( 1 + 5 ) + ( 52 + 53 ) + ... + ( 5403 + 5404 )
= 6 + 52 . ( 1 + 5 ) + ... + 5403 . ( 1 + 5 )
=6 + 52 . 6 + ... + 5403 . 6
= 6 . ( 1 + 52 + ... + 5403 )
= 3 . 2 . ( 1 + 52 + .... + 5403 ) chia hét cho 3
Ta có: 1 + 5 + 52 + .... + 5404
= ( 1 + 5 ) + ( 52 + 53 ) + ... + ( 5403 + 5404 )
= 6 + 52 . ( 1 + 5 ) + ... + 5403 . ( 1 + 5 )
=6 + 52 . 6 + ... + 5403 . 6
= 6 . ( 1 + 52 + ... + 5403 ) chia hết cho 3