\(E=7+7^3+7^5+7^7+...+7^{49}+7^{51}\)
\(=\left(7+7^3\right)+\left(7^5+7^7\right)+...+\left(7^{49}+7^{51}\right)\)
\(=7\left(1+7^2\right)+7^5\left(1+7^2\right)+...+7^{49}\left(1+7^2\right)\)
\(=50\left(7+7^5+...+7^{49}\right)=5\cdot10\cdot\left(7+7^5+...+7^{49}\right)⋮10\)
\(F=1+3+3^2+...+3^{99}+3^{100}+3^{101}\)
\(=\left(1+3+3^2+3^3+3^4+3^5\right)+\left(3^6+3^7+3^8+3^9+3^{10}+3^{11}\right)+...+\left(3^{96}+3^{97}+3^{98}+3^{99}+3^{100}+3^{101}\right)\)
\(=\left(1+3+3^2+3^3+3^4+3^5\right)+3^6\left(1+3+3^2+3^3+3^4+3^5\right)+...+3^{96}\left(1+3+3^2+3^3+3^4+3^5\right)\)
\(=364\left(1+3^6+...+3^{96}\right)\)
\(=7\cdot52\left(1+3^6+...+3^{96}\right)⋮7\)
E=7+73+75+77+...+749+751�=7+73+75+77+...+749+751
=(7+73)+(75+77)+...+(749+751)=(7+73)+(75+77)+...+(749+751)
=7(1+72)+75(1+72)+...+749(1+72)=7(1+72)+75(1+72)+...+749(1+72)
=50(7+75+...+749)=5⋅10⋅(7+75+...+749)⋮10=50(7+75+...+749)=5⋅10⋅(7+75+...+749)⋮10
F=1+3+32+...+399+3100+3101�=1+3+32+...+399+3100+3101
=(1+3+32+33+34+35)+(36+37+38+39+310+311)+...+(396+397+398+399+3100+3101)=(1+3+32+33+34+35)+(36+37+38+39+310+311)+...+(396+397+398+399+3100+3101)
=(1+3+32+33+34+35)+36(1+3+32+33+34+35)+...+396(1+3+32+33+34+35)=(1+3+32+33+34+35)+36(1+3+32+33+34+35)+...+396(1+3+32+33+34+35)
=364(1+36+...+396)=364(1+36+...+396)
=7⋅52(1+36+...+396)⋮7