C/m chia hết cho 3:
2^1+2^2+2^3+...+2^2010
=(2^1+2^1)+(2^3+2^4)+...+(2^2009+2^2010)
=2(1+2)+2^3(1+2)+2^5(1+2)+...+2^2009(1+2)
=(2+2^3+2^5+...+2^2009)(1+2)
=(2+2^3+...+2^2009).3 chia hết cho 3 (1)
C/m chia hết cho 7:
2^1+2^2+2^3+...+2^2010
=(2^1+2^2+3^3)+(2^4+2^5+2^6)+...+(2^2008+2^2009+2^2010)
=2(1+2+2^2)+2^4(1+2+2^2)+...+2^2008(1+2+2^2)
=(2+2^4+2^7+...+2^2008)(1+2+2^2)
=(2+2^4+2^7+...+2^2008).7 chia hết cho 7 (2)
Từ (1)(2) ==> đfcm