# On Relatively Prime Subsets and Supersets

Research paper by **Mohamed El Bachraoui**

Indexed on: **24 Oct '09**Published on: **24 Oct '09**Published in: **Mathematics - Number Theory**

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#### Abstract

A nonempty finite set of positive integers A is relatively prime if gcd(A) =
1 and it is relatively prime to n if gcd(A [ fng) = 1. The number of nonempty
subsets of A which are relatively prime to n is \Phi(A, n) and the number of
such subsets of cardinality k is \Phi_k(A, n). Given positive integers l1, l2,
m2, and n such that l1 <= l2 <= m2 we give \Phi([1;m1][[l2;m2]; n) along with
Phi_k([1;m1] [ [l2;m2]; n). Given positive integers l;m, and n such that l <= m
we count for any subset A of {l,l+1,...,m} the number of its supersets in [l;m]
which are relatively prime and we count the number of such supersets which are
relatively prime to n. Formulas are also obtained for corresponding supersets
having fixed cardinalities. Intermediate consequences include a formula for the
number of relatively prime sets with a nonempty intersection with some fixed
set of positive integers.