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This paper is devoted to the study of the arithmetic graph of a composite number <i>m</i>, denoted by <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="script">A</mi> <mi>m</mi> </msub> </semantics> </math> </inline-formula>. It has been observed that there exist different composite numbers for which the arithmetic graphs are isomorphic. It is proved that the maximum distance between any two vertices of <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="script">A</mi> <mi>m</mi> </msub> </semantics> </math> </inline-formula> is two or three. Conditions under which the vertices have the same degrees and neighborhoods have also been identified. Symmetric behavior of the vertices lead to the study of the metric dimension of <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="script">A</mi> <mi>m</mi> </msub> </semantics> </math> </inline-formula> which gives minimum cardinality of vertices to distinguish all vertices in the graph. We give exact formulae for the metric dimension of <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="script">A</mi> <mi>m</mi> </msub> </semantics> </math> </inline-formula>, when <i>m</i> has exactly two distinct prime divisors. Moreover, we give bounds on the metric dimension of <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="script">A</mi> <mi>m</mi> </msub> </semantics> </math> </inline-formula>, when <i>m</i> has at least three distinct prime divisors.