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The acidity of DNA is caused by the presence of the phosphate groups which are themselves acidic. These consist of a sugar bound to one of the 4 nucleobases Adenine, Cytosine, Guanine or Thymine (Uracil in the case of RNA) and a phosphate group. Despite the fact that DNA does contain many basic groups, their basic properties are masked somewhat because of the fact that they hydrogen bond with each other to form base pairs. Hence it's the acidic part of the molecule that dominates, and that is why we know DNA as an acid.
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