# Somewhat Big Numbers

Number Theory Level 5

A sequence of pairs of integers $$(x_0,y_0),\ (x_1,y_1),\ (x_2,y_2),...$$ is defined, starting from some initial pair $$(x_0,y_0),$$ by the following formulas: For all positive integers $$k$$, $\begin{cases} x_{k}=x_{k-1}^2y_{k-1}\\ y_{k}=x_{k-1}^2y_{k-1}+x_{k-1}y_{k-1}^2. \end{cases}$ Find the smallest $$N$$ such that $$x_N$$ is divisible by $$100!$$ for every initial pair of starting values $$(x_0,y_0).$$

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