Question: I'm currently a high school student trying to build an ultralight aircraft from scratch (I'm self taught and still learning from educational videos), but I don't have much savings (about RM 2.5k, which converts to roughly 587 USD), so I could only afford a small engine. 

Is it possible to achieve a certain propeller's efficient rpm by using gears while still using a small engine? Like those gearboxes found in cars to increase rpm.

Answer: 
The short answer is yes you can use some cheaply available engines to achieve flight however you are presumably asking about things like small car engines or motorcycle engines etc. The issue you have here is that these engines are [inherently different](https://aviation.stackexchange.com/questions/11527/what-are-some-of-the-differences-between-piston-engines-used-in-aircraft-and-aut/11528#11528) from a use case standpoint than your typical aircraft engine. Generally speaking, most car/motorcycle engines rev too high to be used in an aircraft, you actually want to lower the RPM, not raise it. You can read up on it [here](http://www.flyingmag.com/aircraft/do-car-engines-make-good-airplane-engines) and [here](https://www.reddit.com/r/askscience/comments/2fedpn/why_dont_propeller_driven_aircraft_use_automotive/) and [here](https://aviation.stackexchange.com/questions/11527/what-are-some-of-the-differences-between-piston-engines-used-in-aircraft-and-aut). You can pull it off with a stepper gear box ([like a turbo prop uses](https://aviation.stackexchange.com/questions/13837/given-the-same-engine-why-install-a-gearbox-on-a-turboprop-but-not-on-a-turbofa)) and it was even been done on [a piston plane back in the 80's by Mooney](https://en.wikipedia.org/wiki/Porsche_PFM_3200). The other problem you will have with car engines is cooling them, generally cars are watercooled, something that adds a lot of weight and is not ideal for flight. A two stroke or other similar air cooled motorcycle/scooter engine may be a better thing to use. 

Many small engines will produce more than enough horsepower to achieve flight. Keep in mind the [early piper cubs only had 40HP](https://en.wikipedia.org/wiki/Piper_J-3_Cub) which can be easily found in many 2 stroke engines. 

Your best bet may be to look for a [used VW bug engine](http://www.greatplainsas.com/) as they are very similar to aircraft engines and even [were used in some air frames](http://www.enginehistory.org/HOAE/Limbach.html) at one point in history. Due to the cars high production numbers the engines are fairly easy to come by.

You did not list a jurisdiction in your question but the regulations and legislation on this change from place to place, so you may need to see what exactly you are allowed to do in your jurisdiction in relation to home built and experimental aircraft.
Gliders fly without an engine and they do well :) Where I come from, it is common (and legal) to use an old car engine and build it into your own UL plane. Consider getting a car that is very old (70's or 80's) - these can be usually find either for free (after a rear crash) or very very cheap.

These old engines are also easy to remove and fit into an aircraft. In Europe, old Trabant engines are very common for this, but I am sure each region has some car model that is old, unused and waiting in planty in old scrapeyards.

If your legislation allows building a non-certified motor into your plane, than this wold be the way to go. These engines can easily output 40 to 80 kWs of power which is way more than you need.

*Safety notice: If you do this, the #1 rule is: Always remember the engine is non-certified and old, so always fly it like it should break the next minute - avoid flying over large forests and water areas.*