\(A=1+(5+5^2+5^3)+(5^4+5^5+5^6)+(5^7+5^8+5^9)\)
\(\Leftrightarrow A=1+5.\left(1+5+5^2\right)+5^4.\left(1+5+5^2\right)+5^7.\left(1+5+5^2\right)\)
\(\Leftrightarrow A=1+5.31+5^4.31+5^7.31\)
\(\Leftrightarrow A=1+31.\left(5+5^4+5^7\right)\)
Vì \(31.\left(5+5^4+5^7\right)⋮31\)nên A chia cho 31 dư 1.
1 + 5 + 52 + 53 + 54 + 55+ 56+ 57+ 58+ 59 cho 31
=1+( 5 + 52 + 53)+(54 + 55+ 56)+(57+ 58+ 59)
=5.(1+5+52)+54(1+5+52)+57(1+5+52)+1
=1+5. 31+54. 31+57.+31
=31.(5+54+57)+1
Vì 31 chia hết cho 31
Nên 31.(5+54+57) chia hết cho 31
Vì thế 31.(5+54+57) chia cho 31 +1
Vậy tổng này chia 31 dư1
