Looking for help with a "random shut down" problem with a Toshiba Satellite A305

If you can help me solve my problem I will be more than happy to compensate you for your time! (PayPal)

Okay so to be precise, it is a Toshiba Satellite A305-S6857

The problem is that the computer will completely shut off at any given time. It will sometimes boot into windows but it will be a black screen with only the cursor until a blue screen pops up that I never have enough time to read through. It won't stay on trying to install a new OS or in the Setup Manager.
What I've tried:
Removing hardware one by one to see if it was the problem
Completely cleaned heatsink
Removed battery and ran only with power cord
Power drain
Replaced motherboard with a used but working one


Let me know if you need anymore information.
Thank you in advance for your time!

The blue screen should produce a minidump file. Locate this file on your pc and upload it please.

Usually in c/windows/minidump.

How do I go about accessing that?

Go to my computer, then navigate to (c/windows/minidump

There should be a few crash dump files in there, choose the most recent by date and upload it here.

Alternatively, download blue screen view here: http://www.softpedia.com/get/System/Back-Up-and-Recovery/BlueScreenView.shtml

Use it to analyse your minidumps, and google the results it gives you to help diagnose why your computer blue screened.

I can't get into that, the computer won't completely open Windows. Its just a black screen with the cursor.
Same scenario with Safe Mode.