When the taxpayers or a foundation's donors ask how their money has been spent, funding agencies need accurate information for accountability. If continued funding of research is important to you, then citing the grant is good practice. Cite the grant whether your research was directly funded or you received indirect support through services subsidized in a funded Core or Center.
Organization identifiers for your funder, like GRID or RORs, are usually recognized "automagically" if you cite the Grant ID number.
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