Do I have a memory problem or a motherboard problem

This is my first post, so please, be kind. I am running a Toshiba Satellite L675 laptop. Recently (within the last two weeks) my computer has been freezing up. To the point that nothing will work except the cursor. The first time this happened I let the computer sit for an hour and it remained frozen. At that point I did a hard re-start and lo and behold the system failed to boot. I tried re-booting for a good fifteen minutes and all the while it failed to detect my hard drive. So, my initial thought was that my hard drive was failing. I eventually managed to get the computer booted and backed up my files and OS to an external drive. The system ran for another four days until it again froze up and would not re-boot. Again, a good fifteen minutes later it finally remembered that there is a hard drive installed and booted up. This was the last straw, so I have run a hard drive check (HD tune). I ran one with the hard drive removed from my system (actually, a buddy with more toys and knowledge than I have did this for me) and a test with the hard drive in the system and both times the hard drive passed with no errors logged. Fair enough, so we (my buddy and I) decided to trouble shoot this farther. I ran the stress test program called prime in a RAM intensive test, which causes my computer to crash completely off inside of 25 minutes every time. I also have run MemoryTest 86. This is interesting, at least to me. Regardless of what combination of RAM stick and slot that I use (I've reversed the sticks in the slots and tried each stick, one at a time in each slot on its own) every time I run MemoryTest my computer crashes without registering a single pass or error on the RAM. My cooling fan is on and blowing. Is there anything else here worth checking? I really can't afford a new computer, but being as this is the only one I have and it isn't reliable, I need to know if there is any hope or if my motherboard is cooked. Any help is great. Thanks. Bye the way, I have been monitoring my hardware status through all of this, and while the only thing that I have a sensor for is the HDD temp, it is never going above 40 deg celsius.

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OK, OK, for sake of all the competitor teams… that's just my opinion.
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That you repeatedly mention "15 minutes" waiting before the machine boots up again, makes me question HEAT as an issue.
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When you mention testing with the hard drive out of the case, do you mean that you connected the hard drive as a second drive to a known-good machine and then tested it with HD Tune?
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Sometimes a malfunctioning Battery can produce excessive heat.
Consider removing the Battery and running with just the A/C adaptor power source.

Do you get any difference/better performance with the Battery removed?

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Do you, or you plus your buddy, have the technical know-how, skill, and step-by-step patience to open up that case for purpose of doing a good cleaning of the air intake and exhaust vents?

Please read this excellent comment and tutorial by our Digerati, here: http://forums.whatthetech.com/index.php?showtopic=103649

I don't recommend that you take any steps to work inside the case unless you are thoroughly familiar with the information posted by Digerati and sufficiently skilled to take the task slowly, deliberately, and carefully.
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Before you take any steps to open up the case, let's collect some additional information.

You mention hard drive temperature not increasing above 40° C, which is perfectly acceptable.
How are you monitoring temperature?
How do you know that temperature is not increasing to unacceptable levels as the machine approaches an incident of freeze or shut-down?

Consider using SpeedFan to obtain any other temperature measurements that might be available in that machine.
Post your results in your next reply.
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You have a pretty nice machine with good hardware and not likely many years old.
Is there any reason to suspect that the machine has been subjected to abusive use or conditions.
Dropping or other physical damage to the machine would qualify as abusive conditions.
Extreme and prolonged use of the machine in interactive gaming might qualify as abusive use.
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Let's see if we can get an overview of the machine's performance.

Please Run the PCPitstop.com OverDrive Full Tests

Here's how:

You must use your Internet Explorer for this procedure. (doesn't work so well in Firefox or others). If your machine is running Vista or Windows 7, you must Select IE to “Run as Administrator”. After completing PCPitstop OverDrive you can close your IE browser and re-open it Normally so that you are no longer running as administrator.

Go to: http://www.pcpitstop.com
Click on "Free Computer Check-up" listed below PC Pitstop OverDrive
In the User Login - Click on "Sign up FREE!"
You'll need to submit a valid email address and create your own password, then click - Create Account(button)

Now enter your email address and password to Log in, Select - Scan this system Now!(button)
You will then be asked to download an ActiveX component and allow it to install.
It is safe and does not compromise your privacy.
Follow the on-screen prompts to install the ActiveX and to allow the Full Tests to be run on your machine.

The Full Tests take about 2 1/2 - 3 minutes on most machines.
When complete, a Results - Summary - Recommended Fixes will be displayed.

Please post the URL internet address, from your Results, back here into this Topic Thread so that we can review the configuration and present performance levels of your machine.

Note: During the graphics 2D and graphics 3D testing, your screen will display some rapidly moving objects.
If you are sensitive to visual flashing, it may cause dizziness. Therefore, look away from the screen during that portion of the testing.

After reviewing the results we will be more informed and may be able to provide better recommendations for you to work towards improving your machine's performance.

While PCPitstop does offer a variety of Paid Products, the PCPitstop OverDrive testing is FREE. Please ignore the references to Paid Products. At WTT we prefer to provide manual solution instructions that you can apply directly to your machine.

Best Regards