cmi-entrance 2019 QA1

cmi-entrance · India · ugmath 4 marks Number Theory Divisibility and Divisor Analysis

For a natural number $m$, define $\Phi_{1}(m)$ to be the number of divisors of $m$ and for $k \geq 2$ define $\Phi_{k}(m) := \Phi_{1}\left(\Phi_{k-1}(m)\right)$. For example, $\Phi_{2}(12) = \Phi_{1}(6) = 4$. Find the minimum $k$ such that $$\Phi_{k}\left(2019^{2019}\right) = 2.$$

For a natural number $m$, define $\Phi_{1}(m)$ to be the number of divisors of $m$ and for $k \geq 2$ define $\Phi_{k}(m) := \Phi_{1}\left(\Phi_{k-1}(m)\right)$. For example, $\Phi_{2}(12) = \Phi_{1}(6) = 4$. Find the minimum $k$ such that
$$\Phi_{k}\left(2019^{2019}\right) = 2.$$

Paper Questions

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