\(S=5+5^2+5^3+.......+5^{2003}\)
\(\Leftrightarrow S=5+\left(5^2+5^3+5^4+5^5+5^6+5^7\right)+........+\left(+5^{1998}+5^{1999}+5^{2000}+5^{2001}+5^{2002}+5^{2003}\right)\)
\(\Leftrightarrow S=5+5^2\left(1+5+5^2+5^3+5^4+5^5\right)+....+5^{1998}\left(1+5+5^2+5^3+5^4+5^5\right)\)
\(\Leftrightarrow S=5+5^2.3906+.......+5^{1998}.3906\)
\(\Leftrightarrow S=5+3906\left(5^2+......+5^{1998}\right)\)
Mà \(3906\left(5^2+....+5^{1998}\right)⋮126\)
\(\Leftrightarrow5+3906\left(5^2+.....+5^{1998}\right)\) chia 126 dư 5
b/ \(S=5+5^2+........+5^{2003}\)
\(=5+\left(5^2+5^3\right)+\left(5^4+5^5\right)+.......+\left(5^{2002}+5^{2003}\right)\)
\(=5+5^2\left(5+5^2\right)+.......+5^{2002}\left(5+5^2\right)\)
\(=5+5^2.30+........+5^{2002}.30\)
\(=5+30\left(5^2+.....+5^{2002}\right)\)
Mà \(30\left(5^2+......+5^{2002}\right)⋮10\)
\(\Leftrightarrow5+30\left(5^2+......+5^{2002}\right)\) chia 10 dư 5
\(\Leftrightarrow S\) có chữ số tận cùng là 5
S=5+52+53+.......+52003S=5+52+53+.......+52003
⇔S=5+(52+53+54+55+56+57)+........+(+51998+51999+52000+52001+52002+52003)⇔S=5+(52+53+54+55+56+57)+........+(+51998+51999+52000+52001+52002+52003)
⇔S=5+52(1+5+52+53+54+55)+....+51998(1+5+52+53+54+55)⇔S=5+52(1+5+52+53+54+55)+....+51998(1+5+52+53+54+55)
⇔S=5+52.3906+.......+51998.3906⇔S=5+52.3906+.......+51998.3906
⇔S=5+3906(52+......+51998)⇔S=5+3906(52+......+51998)
Mà 3906(52+....+51998)⋮1263906(52+....+51998)⋮126
⇔5+3906(52+.....+51998)⇔5+3906(52+.....+51998) chia 126 dư 5
b/ S=5+52+........+52003S=5+52+........+52003
=5+(52+53)+(54+55)+.......+(52002+52003)=5+(52+53)+(54+55)+.......+(52002+52003)
=5+52(5+52)+.......+52002(5+52)=5+52(5+52)+.......+52002(5+52)
=5+52.30+........+52002.30=5+52.30+........+52002.30
=5+30(52+.....+52002)=5+30(52+.....+52002)
Mà 30(52+......+52002)⋮1030(52+......+52002)⋮10
⇔5+30(52+......+52002)⇔5+30(52+......+52002) chia 10 dư 5
⇔S⇔S có chữ số tận cùng là 5