\(A=3+3^2+3^3+3^4+...+3^{90}\)
\(=\left(3+3^2+3^3+3^4+3^5\right)+...+\left(3^{86}+3^{87}+3^{88}+3^{89}+3^{90}\right)\)
\(=3.\left(1+3+3^2+3^3+3^4\right)+...+3^{86}\left(1+3+3^2+3^3+3^4\right)\)
\(=3.121+...+3^{36}.121\)
\(=121\left(3+...+3^{86}\right)⋮11\left(dpcm\right)\)
\(A=3+3^2+3^3+3^4+...+3^{90}\)
\(=\left(3+3^2+3^3\right)+\left(3^4+3^5+3^6\right)+...+\left(3^{88}+3^{89}+3^{90}\right)\)
\(=\left(3+3^2+3^3\right)+\left(3^3.3+3^3.3^2+3^3.3^3\right)+...+\left(3^{87}.3+3^{87}.3^2+3^{87}.3\right)\)
\(=\left(3+3^2+3^3\right)+3^3\left(3+3^2+3^3\right)+...+3^{87}\left(3+3^2+3^3\right)\)
\(=39.1+3^3.39+...3^{87}.39\)
\(=39\left(3^3+1+...+3^{87}\right)\)
\(=13.3\left(3^3+1+...+3^{87}\right)⋮13\left(dpcm\right)\)
