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.
Actually, you'd be gearing the engine down, not up. Propellers become very inefficient if the tips go supersonic (as well as extremely noisy). You'd want to keep the prop at around 1800-2200 rpm, depending on the size of the prop. Smaller engines typically put out peak power at much higher rpm's than that, so they have to be geared down to match the prop to the engine's power curve. 

A typical ultralight engine is the [Rotax 582](https://www.flyrotax.com/produkte/detail/rotax-582-ul.html). Designed for ultralight aircraft use, it can sustain high power output for long periods of time. That is current production, so it can be a bit expensive.

Now out of production but they can be found on the used market fairly inexpensively, is the [Rotax 503](https://en.wikipedia.org/wiki/Rotax_503). It has been a very popular ultralight engine. 

Motorcycle engines may have less than desirable reliability. Typically, they are designed for brief periods of acceleration, with long periods of cruising at low power output. If you take a stock small motorcycle engine and run it wide open for long periods of time, the cylinder head will probably overheat, blowing the gasket and leaving you with no power. When that happens, you are going to come down, regardless of whether there is a safe place to land, or not.

I have seen Rotax 503's with a gearbox for ultralight use, in the USD 1500 range. More than a small motorcycle engine, but cheaper than a funeral...