How to root the Galaxy S3 4G LTE GT-i9305 on Android Jelly Bean 4.1.2

This tutorial is exclusively for the Samsung Galaxy S3 4G LTE GT-i9305 model running on Android Jelly Bean 4.1.2 XXBLL5. If you try to flash another model or Android version with this root, you could break your smartphone.
Neither AndroidPIT nor I will be held responsible for any damages or losses of data. Make sure you have completed a full backup of your files before starting. Not sure what rooting/flashing/ROMS even are? Check out our Rooting for Beginners Q&A's and some custom ROMS that may be at your level.For check this search in this site.

Again, do NOT try this on your Galaxy S3 i9300!!!!!

Prerequisites

1. Activate the USB debugging mode on your S3. To do this, simply go to Settings → Developer options → USB debugging.
2. Make sure that the drivers for your Galaxy S3 have been installed on your computer. In order to do this, connect your smartphone to your PC with your micro USB cord and wait until your smartphone is recognized by Windows.
3. Charge your smartphone until at least 75% full.
4. Download and decompress Odin on your computer.
5. Download the file called I9305_CWM_Recovery.tar.md5 
6. Download the file called CWM-SuperSU-v0.96.zip

Install Clockwork Recovery (CWM)

1. Launch Odin
2. Place your Galaxy S3 4G in download mode (volume down+home+power for about 10 seconds)
3. Connect your Samsung Galaxy S3 4G LTE GT-i9305 via the USB cable to your PC and wait until it's recognized by Odin. A small rectangular box in the top left of the window showing O:COM should appear.
4. Click on PDA and select the I9305_CWM_Recovery.tar.md5 file, which you have downloaded in step 5 of the prerequisites.
5. In Odin, don't touch any other options and make sure that no additional boxes have been checked.
6. Click on start, and wait until the Galaxy S3 reboots. Power it down and up one more time.

The first part is done!

The Root

1. Start off by taking advantage of the Custom Recovery that you have just installed by completing a full backup.
2. To do this, restart your smartphone in recovery mode (volume up+home+power for 10 seconds)
3. You will then navigate within the menu using the volume buttons and select ''backup and restore'', and then ''backup''.
4. Then select ''Reboot now''
5. Place the file CWM-SuperSU-v0.96.zip on the root of your phone.
6. Restart your smartphone again in recovery mode (see step 1) and select ''install zip from SD card''.
7. Navigate to the CWM-SuperSU-v0.96.zip file and validate.
8. Once the file has been installed, return to the main menu and select reboot now.

If this doesn't work on your phone

1. If you aren’t able to get access to the Custom Recovery, this means that the original Android recovery mode re-installed itself when powering up and crushed the Custom Recovery. Follow these instructions to bypass this problem.
2. Complete the same procedure to install the CWM but at the end when the Samsung logo appears when powering up the S3 4G, wait 2 seconds and turn it off right away. The image following the Samsung logo shouldn't have the time to load.
3. Next, proceed with the root.
4. Then, install a file explorer (File Explorer, ES File Explorer File Manager etc) with the root rights and search for the file named boot-recovery.p or else recover-from-boot which can be found in the directory /system. Rename it by adding .bak at the end (example: boot-recovery.p-bak). Next, re-do the installation process for Clockwork Recovery.

That should do it!