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On Relatively Prime Subsets, Combinatorial Identities, and Diophantine Equations
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Mohamed El Bachraoui

Department of Mathematical Sciences

United Arab Emirates University

P.O. Box 17551

Al-Ain

United Arab Emirates

**Abstract:**

Let *n* be a positive integer and let *A* be a nonempty finite set of
positive integers. We say that *A* is relatively prime if
gcd(*A*) = 1,
and that *A* is relatively prime to *n* if gcd(*A*,*n*)=1.
In this work
we count the number of nonempty subsets of *A* that are relatively
prime and the number of nonempty subsets of *A* that are relatively
prime to *n*. Related formulas are also obtained for the number of such
subsets having some fixed cardinality. This extends previous work for
the case where *A* is an interval of successive integers. As an
application we give some identities involving Möbius and Mertens
functions, which provide solutions to certain Diophantine equations.

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Received September 23 2011;
revised version received February 2 2012; March 14 2012.
Published in *Journal of Integer Sequences*, March 25 2012.

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