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Is there a country that uses the term “dollar” for currency without also using “cents” as fractional monetary units?
Wikipedia has a nice list of currencies that use "cents" and currencies that use 1/100th division that is not called "cent". Cent means "100" in Latin (and French, and probably all the Roman family of languages), so if the currency is divided by 100 subunits - it will likely to be called "cent" or something similar in the local language. The list of currencies (on the same page) where it is not the case is significantly shorter, and includes countries with relatively ancient currency units that were invented before the introduction of the decimal system (even though now they are in fact decimal they still kept the old names, like the British "pence" or the Russian "kopek"). The point is that "Dollar" and "cent" are not directly related, many currencies that are not called "Dollar" are using cents as well (Euro, among others). It just means "1/100th", and it is safe to assume that most (if not all) of the modern currencies are divided into 1/100th.