# Arithmetic Integral

Number Theory Level 5

$\Large \displaystyle\int n\text{ }dx \neq nx?$

In calculus, when you take the derivative of a constant you get zero as an answer. In number theory, there is something called the arithmetic derivative which allows you to differentiate a number and get a nonzero answer. The arithmetic derivative works as follows.

Where $$n'$$ denotes the arithmetic derivative of $$n$$:

$$p' = 1$$ for all primes $$p$$

$$(ab)'=a'b+ab'$$

$$0'=1'=0$$

For example, $$6'=(2\times3)'=(2')(3)+(2)(3')=(1)(3)+(2)(1)=5$$

While you clearly get a single answer when taking the arithmetic derivative of a number, multiple values of $$n$$ can lead to the same $$n'$$. Let us define the arithmetic integral, denoted $$\int n$$, as the function that when given a value of $$n$$ returns a set of all the positive integers $$m$$ such that $$m'=n$$. What is the value of $$n$$, where $$1<n<100$$, that gives us the set with the largest dimensions? In other words, what is the value of $$n$$ that has the most solutions $$m$$ to the equation $$m'=n$$?

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