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You don't hook up a sprocket to a disc brake! If you mean how to mount the rear gear cluster (freewheel) on a rear wheel, a special tool is required to remove the old one, then you just screw on the new one by hand. If you mean something else then you need to explain more clearly or provide some pics.
I am sorry I could not find a manual for your Mongoose. I will suggest you go to the following website http://www.youtube.com/watch?v=2j1gSwhsVN8 You can find numerous videos for your bike repair. in Youtube Bud
To remove any rear freewheel or cassette you will need to purchase a Park Tool. Park Tool is the best brand when it comes to bike repair. Google Park Tool and you can look through their products. You will need a cassette/freewheel remover. There are dozens of different companies that make hubs and cassettes (gear sprocket). Take a look at yours and see what company made it. You can than find out which tool with properly fit your cassette. These tools are made to fit in a wrench making hand removal easy. Some brands will require a chain whip. The only way you will know which too you need is by find the company name on the cassette/hub. Without this tool removal is virtually impossible. If you need help narrowing down which cassette remover you need you can always let me know the company name and I will let you know the proper cassette tool to purchase.
Theres a special wrench you need that fits the gear. Its easiest to take the wheel to a shop that has one and they will also be able to replace it with one with the teeth ratio you'll want.
I think I can help. Place one end of the chain over the rear gear (doesn't matter which one, the biggest) and adjust the lever or twist ring on the handlebar so the derailleur lines up in a straight line with it. Pull the chain from the rear sprocket down and over the pivot wheel and back over the bottom wheel so it looks just like this S and the end comes forward. Join the chain links. Place the chain over a couple front teeth and turn the pedal carefully until the chain seats on the front sprocket.
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