2120.12 – Factor!

Factor the number 612+88+2786^{12} + 8^8 + 27^8 without using 11 as a factor.

Solution

612+88+278=(2⋅3)12+(23)8+(33)8=212⋅312+224+324=224+212⋅312+324=224+2(212⋅312)+324−212⋅312=(212)2+2(212⋅312)+(312)2−212⋅312=(212+312)2−(26⋅36)2=(212+312+26⋅36)(212+312−26⋅36)=582193⋅488881\begin{aligned}6^{12} + 8^8 + 27^8 &= (2 \cdot 3)^{12} + (2^3)^8 + (3^3)^8 \\[0.5em]&= 2^{12} \cdot 3^{12} + 2^{24} + 3^{24} \\[0.5em]&= 2^{24} + 2^{12} \cdot 3^{12} + 3^{24} \\[0.5em]&= 2^{24} + 2 (2^{12} \cdot 3^{12}) + 3^{24} - 2^{12} \cdot 3^{12} \\[0.5em]&= (2^{12})^2 + 2 (2^{12} \cdot 3^{12}) + (3^{12})^2 - 2^{12} \cdot 3^{12} \\[0.5em]&= (2^{12} + 3^{12})^2 - (2^6 \cdot 3^6)^2 \\[0.5em]&= (2^{12} + 3^{12} + 2^6 \cdot 3^6) (2^{12} + 3^{12} - 2^6 \cdot 3^6) \\[0.5em]&= 582193 \cdot 488881\end{aligned}