Ta có:
\(\left(2^3+1\right)\left(3^3+1\right)...\left(100^3+1\right)\)
\(=\left(2+1\right)\left(4-2+1\right)\left(3+1\right)\left(9-3+1\right)...\left(100+1\right)\left(100^2-100+1\right)\)
\(=3.3.4.7...101.9901\)
\(=\left(3.4.5...101\right)\left(3.7.13...9901\right)\)
\(\left(2^3-1\right)\left(3^3-1\right)...\left(100^3-1\right)\)
\(=\left(2-1\right)\left(4+2+1\right)\left(3-1\right)\left(9+3+1\right)...\left(100-1\right)\left(100^2+100+1\right)\)
\(=1.7.2.13.3.21...99.10101\)
\(=\left(1.2.3...99\right)\left(7.13.21.10101\right)\)
=> \(\frac{\left(2^3+1\right)\left(3^3+1\right)...\left(100^3+1\right)}{\left(2^3-1\right)\left(3^3-1\right)...\left(100^3-1\right)}\)
\(=\frac{\left(3.4.5...101\right)\left(3.7.13...9901\right)}{\left(1.2.3...99\right)\left(7.13.21.10101\right)}=\frac{\left(100.101\right).3}{\left(1.2\right).10101}=\frac{30300}{20202}\)
\(=3.3.4.7...101.9901\)