# Coprime Period

$\large\underbrace{0,1,0,1,0,0,0,1,0,1}_{\text{period}}, 0,1,0,1,0,0,0,1,0,1,0,\dots$ Let $$n$$ be a positive integer, and let the function $$f_n:\ \mathbb N \to \{0,1\}$$ be defined by $f_n(m) = \begin{cases} 0 & \gcd(n,m) > 1 \\ 1 & \gcd(n,m) = 1. \end{cases}$ That is, $$f_n(m)$$ tells us whether $$m$$ is coprime to $$n$$ or not. If we write out the values of $$f_n,$$ we get a periodic (repeating) pattern. For instance, the list above gives the values of $$f_{10}$$ starting at $$m = 0$$; the pattern repeats itself with period 10.

Consider the function $$f_{1400}$$. What is its period?


Note: The period is defined as the smallest possible value, that is, the least positive $$T$$ such that $$f_n(m + T) = f_n(m)$$ for all integers $$m$$.

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