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Let \( G \) be a \((p,q)\)-graph where each edge of \( G \) is labeled by a number \( 1, 2, \ldots, q \) without repetition. The vertex sum for a vertex \( v \) is the sum of the labels of edges that are incident to \( v \). If the vertex sums are equal to a constant (mod \( k \)) where \( k \geq 2 \), then \( G \) is said to be Mod(\( k \))-edge-magic. In this paper, we investigate graphs which are Mod(\( k \))-edge-magic. When \( k = p \), the corresponding Mod(\( p \))-edge-magic graph is the edge-magic graph introduced by Lee (third author), Seah, and Tan in \([10]\). In this work, we investigate trees, unicyclic graphs, and \((p, p+1)\)-graphs which are Mod(2)-edge-magic.