# 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 \(m'=n\)?