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the states and their city/town governments pay for all elections
In addition to the US Federal Government and the 50 state governments, there are county, city and town governments in every state. So the total number of governments in the US can be counted in the thousands.
The Constitution of the United States, for the most part, has described the role of the Federal government, the basic rights of US citizens, and the limits of the Federal governments powers. As such, it's my opinion that it is a current and thus primary source of information.
State governments have some powers that the federal government does not have.
The main role of most local governments in the US is to the public. Local governments manage things such as school districts, public transportation, libraries and sewer/water systems.
Residents of US territories, such as Puerto Rico, Guam, and the US Virgin Islands, cannot vote in federal elections. However, they are allowed to vote in their respective local elections and have limited representation in the US Congress through non-voting delegates.
how often do presidential elections take place
The power to conduct elections was left to the states. Each state oversees the federal elections within its borders.
In the US, the Federal government's powers are derived by the US Constitution. Taking this back a step, it was the states that created the Constitution. At its beginning, however, the source of the US Federal governments powers are derived from the citizens of the USA.
By dividing some responsibilities between the federal and state governments
That depends on the constitution in question. In the US it would be federal elections.
18 years of age and citizen of US