Video solution:

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Written solution:

Begin by finding the prime factorization of $23232.$ To do this, we repeatedly factor out the smallest prime factor from the number, a process that terminates when the number is a prime number. This process is outlined below: $$\begin{align*}23232 &= 2\cdot 11616\\ &= 2^2\cdot 5808\\ &= 2^3\cdot 2904\\ &= 2^4\cdot 1452\\ &= 2^5\cdot 726\\ &= 2^6\cdot 363\\ &= 2^6\cdot 3 \cdot 121\\ &=2^6\cdot 3\cdot 11^2 \end{align*}$$ An arbitrary factor of $23232$ can be created by taking the product of any number of prime factors. More explicitly, as $23232$ can be represented $p_1^{e_1} p_2^{e_2} \cdots p_m^{e_m}$ where $p$ is a prime number, each factor has $(e_1+1) \cdots (e_m+1)$ options of prime factorizations to choose from, and thus, there are $(e_1+1) \cdots (e_m+1)$ factors. Plugging in values, we can see that there are $$\begin{align*}(6+1)(1+1)(2+1) &= 7\cdot2\cdot3\\ &= 42 \end{align*}$$ factors of $23232.$

Thus, E is the correct answer.