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B = (1 + 3) + (32+33)+.....+(389+390)
= 4 + 32 .(1 + 3) + .....+390.(1+3)
= 1 .4 + 32.4 + ..... +390.4
= 4.(1 + 32 + .... +390) chia hết cho 4
\(S=3+3^2+3^3+3^4+....+3^{89}+3^{90}\)
\(=\left(3+3^2+3^3\right)+\left(3^4+3^5+3^6\right)+...+\left(3^{88}+3^{89}+3^{90}\right)\)
\(==3\left(1+3+3^2\right)+3^4\left(1+3+3^2\right)+3^{88}\left(1+3+3^2\right)\)
\(=\left(1+3+3^2\right).\left(3+3^4+....+3^{88}\right)\)
\(=13\left(3+3^4+...+3^{88}\right)\)\(⋮\)\(13\)
\(A=3+3^2+3^3+...+3^9\)
\(=\left(3+3^2+3^3\right)+\left(3^4+3^5+3^6\right)+\left(3^7+3^8+3^9\right)\)
\(=3\left(1+3+3^2\right)+3^4\left(1+3+3^2\right)+3^7\left(1+3+3^2\right)\)
\(=\left(1+3+3^2\right)\left(3+3^4+3^7\right)\)
\(=13\left(3+3^4+3^7\right)\)\(⋮\)\(13\)
\(A=3+3^2+3^3+...+3^8+3^9\)
\(A=\left(3+3^2+3^3\right)+\left(3^4+3^5+3^6\right)+...+\left(3^7+3^8+3^9\right)\)
\(A=3\left(1+3+3^2\right)+3^4\left(1+3+3^2\right)+...+3^7\left(1+3+3^2\right)\)
\(A=3\left(1+3+9\right)+3^4\left(1+3+9\right)+...+3^7\left(1+3+9\right)\)
\(A=3\cdot13+3^4\cdot13+...+3^7\cdot13\)
\(A=13\left(3+3^4+...+3^7\right)\)
\(\Rightarrow A⋮13\)
Vậy A\(⋮13\left(đpcm\right)\)