\[\begin{array}{l} 1)\,\,\,Chung\,\,minh:\\ a)\,\,\,\overline {ab} + \overline {ba} \,\,\, \vdots \,\,11\\ Ta\,\,co:\,\,\,\,\,\overline {ab} + \overline {ba} = 10a + b + 10b + a = 11a + 11b = 11\left( {a + b} \right)\,\, \vdots \,\,\,11\\ \Rightarrow \overline {ab} + \overline {ba} \,\,\, \vdots \,\,11.\\ b)\,\,\,\,\overline {ab} - \overline {ba} \,\,\, \vdots \,\,9\\ Ta\,\,co:\,\,\,\,\,\overline {ab} - \overline {ba} = 10a + b - \left( {10b + a} \right) = 9a + 9b = 9\left( {a + b} \right)\,\, \vdots \,\,\,9\\ \Rightarrow \overline {ab} - \overline {ba} \,\,\, \vdots \,\,9.\\ 2)\,\,\,Chung\,\,minh:\\ a)\,\,\,\left( {\overline {ab} + \overline {cd} } \right)\,\,\, \vdots \,\,99 \Rightarrow \overline {abcd} \,\, \vdots \,\,99\\ Ta\,\,\,co:\,\,\,\,\left( {\overline {ab} + \overline {cd} } \right)\,\,\, \vdots \,\,9\,\,\,va\,\,\,\,\left( {\overline {ab} + \overline {cd} } \right)\,\,\, \vdots \,\,11\\ \left( {\overline {ab} + \overline {cd} } \right)\,\,\, \vdots \,\,9 \Rightarrow \left( {a + b + c + d} \right)\,\, \vdots \,\,9\\ \Rightarrow \overline {abcd} \,\, \vdots \,\,9.\\ \left( {\overline {ab} + \overline {cd} } \right)\,\,\, \vdots \,\,11 \Rightarrow \left( {10a + b + 10c + d} \right)\,\, \vdots \,\,11\\ \Rightarrow 10\left( {a + c} \right) + \left( {b + d} \right)\,\, \vdots \,\,11\\ \overline {abcd} = 1000a + 100b + 10c + d = 10\left( {a + c} \right) + 990a + 99b + \left( {b + d} \right)\\ = \left[ {100\left( {a + c} \right) + \left( {b + d} \right)} \right] + 11\left( {90a + 9b} \right)\\ Vi\,\,\,10\left( {a + c} \right) + \left( {b + d} \right)\,\, \vdots \,\,11\,\,va\,\,\,11\left( {90a + 9b} \right)\, \vdots \,\,11\\ \Rightarrow \,\overline {abcd} \,\,\, \vdots \,\,11\\ \Rightarrow \overline {abcd} \,\, \vdots \,\,9\,\,\,va\,\,\,\overline {abcd} \,\, \vdots \,\,11 \Rightarrow \overline {abcd} \,\, \vdots \,\,99.\\ Vay\,\,neu\,\left( {\overline {ab} + \overline {cd} } \right)\,\,\, \vdots \,\,99\,\,\,thi\,\,\overline {abcd} \,\, \vdots \,\,99\,.\\ Cau\,\,b\,\,\,chung\,\,minh\,\,\,tuong\,\,tu.\\ Cau\,\,\,3:\\ a)\,\,\,A = 1 + 3 + {3^2} + .... + {3^{1998}} + {3^{1999}} + {3^{2000}}\,\,\, \vdots \,\,13\\ So\,\,\,cac\,\,\,so\,\,\,hang\,\,cua\,\,\,A\,\,\,la:\,\,\,\,2000 - 0 + 1 = 2001\,\,\,so\,\,hang.\\ \Rightarrow Nhom\,\,\,A\,\,duoc\,\,\,667\,\,nhom,\,\,\,moi\,\,nhom\,\,\,co\,\,\,3\,\,\,so\,\,hang.\\ \Rightarrow A = \left( {1 + 3 + {3^2}} \right) + \left( {{3^3} + {3^4} + {3^5}} \right) + .... + \left( {{3^{1998}} + {3^{1999}} + {3^{2000}}} \right)\\ = \left( {1 + 3 + {3^2}} \right) + {3^3}\left( {1 + 3 + {3^2}} \right) + ...... + {3^{1998}}\left( {1 + 3 + {3^2}} \right)\\ = \left( {1 + 3 + {3^2}} \right)\left( {1 + {3^3} + ..... + {3^{1998}}} \right)\\ = 13\left( {1 + {3^3} + .... + {3^{1998}}} \right)\\ \Rightarrow A\,\,\, \vdots \,\,\,13.\\ Cau\,\,b\,\,\,lam\,\,\,tuong\,\,tu. \end{array}\]