Exercise 1. Calculate

$$
\frac{1 \times 2 \times 4+2 \times 4 \times 8+3 \times 6 \times 12+4 \times 8 \times 16}{1 \times 3 \times 9+2 \times 6 \times 18+3 \times 9 \times 27+4 \times 12 \times 36}
$$

Only a numerical answer is expected here. The answer should be given as an irreducible fraction (i.e., in the form $\frac{a}{b}$ where $a$ and $b$ are integers with no common divisors).
Solution to Exercise 1 Instead of calculating the numerator and the denominator, we can notice that the terms of the sum in the numerator are the numbers $i \times(2 i) \times(4 i)=8 i^{3}$ for $1 \leq i \leq 4$ and that the terms of the sum in the denominator are the numbers $i \times(3 i) \times(9 i)=27 i^{3}$ for $1 \leq i \leq 4$. By simplifying the numerator and the denominator by the number $\left(1^{3}+2^{3}+3^{3}+4^{3}\right)$, we obtain the irreducible fraction $\frac{8}{27}$.

The expected answer is therefore $\frac{8}{27}$.