I usually identify each wire in the connectors and where they go before I remove the old engine. With the connectors usually being different shapes you sometimes can remove the wire in the connector by releasing the crimp connector by sliding a small screw driver blade in from the rear and pull it out and just sliding it into the engine connector. if not you can splice, recrimp, or solder your connections.

You will probably need the wiring diagramns from your LT showing the kolher and with wiring diagram from the briggs engine. as for the starter one wire coming from the start position will make the solenoid actuate you should be able to find a wiring diagram online that shows how a starter system is wired by goggling. If you mount it to metal that maybe good enough for a ground.

I will prepare a wiring schematic showing you how to connect your OEM wiring to the new configuration but it will be tomorrow before I can do this. In the mean time, does the starter solenoid you bought have 1, or 2, small terminals (in addition to the two BIG posts)? Also, decide where you want to mount this starter solenoid on the tractor frame, then you will need to purchase a cable the same diameter (wire guage size) as the positive battery cable that will reach from the starter solenoid to the cable post on the starter itself. This new cable must have eyelet terminal ends that fit over the big posts on the starter and the solenoid. Once you decide how long this cable needs to be, you might find an appropriate "ready made" cable at an outdoor power shop, or even an auto parts house.

To reiterate a question from my post yesterday: "does the starter solenoid you bought have 1 small terminal, or 2 small terminals (in addition to the two BIG posts)?"
The answer will be either 1, or it will be 2.

To reiterate a question from my post yesterday: "does the starter solenoid you bought have 1 small terminal, or 2 small terminals (in addition to the two BIG posts)?"
The answer will be either 1, or it will be 2.

Mowie, I have 1 small terminal and the solenoid is grounded. In addition, I found out that it's the white wire that is the ignition wire which needs to go to the 1 small terminal. However, I don't think I am wiring it correctly because when I start the ignition, nothing happens.

I am pretty sure I am not doing something basic. I can't even turn on the head light. When I turn, I don't hear any click sound as well.

In looking at the wiring diagrams for both the machines (917.276020 & 917.276310), I see no differences between the two except for the Kohler version (020) featuring an operator presence relay where the Briggs version (310) features a more "complex" seat switch to accomplish the same goal.
If,........you are connecting the white wire to the small terminal on the starter solenoid, you are making the correct connection,........AND the solenoid is in fact mounted to a grounded, metal surface or has been provided with a separate grounding jumper wire to ground the solenoid case! So, you might want to test that the solenoid is grounded by touching the probe of a test light (or volt meter) to the battery positive post and the other end/lead to the metal case of the solenoid body. If you detect voltage or the light burns, the solenoid is grounded. If you don't detect voltage, it is not grounded and you need to address that first.
So let's go through some checks to determine what the trouble might be. The following outline is a step by step that will guide you through. Refer to the wiring diagram for your 917.276020 for the identifying "alpha characters" I refer to in the text. Make certain there is not a blown fuse before you do these tests.

(begin test outline)
It could be a "bad key switch" but my feeling is "bad clutch/brake switch", but it could be a "bad PTO switch" just as easy, so let's do some testing first. You have a volt meter and/or 12V test light? Be advised that moving or wiggling wires while doing these tests could result in continuity being accidently restored AND this might cause the starter to engage when you don't expect it to. To guard against this, unplug the small white wire from the starter solenoid and cover the wire terminal with tape to prevent a short. First off, you will need to access the backside of the key switch. Step 1. Find the WHITE wire on the back of key switch connector. (leave the connector attached to the switch) While holding the key in the start position, probe the area where the white wire enters the connector body with a test light or a volt meter. You should detect voltage on the test light or the volt meter. If you do not detect voltage, replace the key switch. If you detect voltage, go to step 2. Step 2. Find the PTO switch and identify terminals C & G on the switch. Probe terminal C while the key is in start position. If you detect voltage at C, probe terminal G next. If you detect voltage at C but not at G, replace the PTO switch (make double dang sure the PTO switch is in the OFF position). If you detect voltage at C & G, go to step 3. Step 3. Locate the clutch/brake interlock switch (should be near the pedal or close by, must be attached so pedal linkage can operate the switch. This switch will feature 4 wires, 2 will be white. With the key in the start position, and the pedal locked or held down, probe the white wires of the brake/clutch switch. If you detect voltage on one white wire only, make certain that the switch plunger is actually moving in far enough to operate the switch. If the plunger is moving far enough to operate the switch, but you still detect voltage on just 1 white wire of the switch, replace the clutch/brake switch. The seat switch does not inhibit cranking on this mower, it only functions to kill the ignition spark. Malfunctions in the PTO switch or the clutch/brake switch are capable of killing spark AND inhibiting the starter. Remember to reconnect the white wire to the starter solenoid after testing. I hope this helps in troubleshooting the interlock circuits. (end outline)

After doing these tests, post back your results.

Mownie, solenoid is grounded, but when I went through your circuit test, there were zero voltages on the key switch (white wire) and on the pto terminals so I didn't even bother to test further for clutch/brake. I cannot even turn on the head light. I did both voltage test and the continuity test from the battery to the solenoid and to the starter. The reading below doesn't make sense to me.

[---Label---]
Battery ( -[a] +[b])
Solenoid ([c] [d]) [e]small white wire on solenoid)
[c]-> has additional think red wire
[d]-> connected to the starter
Starter [f]

[--RESULTS---]
1. connected [a][b] -> 12 Volt, but no continuity
2. connected [a] [c] -> 0 Volt, but continuity
3. connected [a] [f] -> 0 Volt, but continity
4. connected a[e] -> 0 Volt, but continuity
5. connected [a][c] -> 12 Volt, but no continuity

I bought a generic solenoid made for MTD from Lowes, but I am not sure whether this is the cause. In addition, I did tighten the electric clutch one revolution (two out of three nuts). Besides this, it's the white wire which is routed to the newly bought solenoid. I've looked at the Kohler engine, and this white wire from the harness connects to the Kohler engine harness which then connects to the Kohler built in solenoid. Within the engine this connects to the ignition coil of the Kohler engine. So what I did was to disconnect it from going to the harness and connected the white wire to the newly bought solenoid. Prior to changing the engine, engine would turn on and off (sometimes) so I did put a tape on the spring of the seat switch which fixed the engine turning on and off.

Again, thank you Mownie for your assistance.

Mownie, I hear no clicking sound as well.

jsw, let's do this using a different strategy.
When you installed the MTD starter solenoid, you would have connected the "Battery Positive" cable to one of the large posts on the MTD solenoid. Was there a second smaller red wire molded into the positive cable at the starter solenoid post on the Kohler starter, or was there a separate smaller red wire that connected to the same starter post along with the battery positive cable?
Either way, the wiring must be configured to the MTD solenoid the same way it was connected to the cable lug on the Kohler starter. Here is why: The small red wire is the wire that feeds battery positive power to the ignition key switch. If this wire was a separate wire, and has not yet been connected to the MTD solenoid on the same post as the battery positive cable, the key switch is essentially "unpowered" at present time. This small wire also is equipped with the "Main Fuse" for the electrical wiring of the tractor. If this fuse is blown, that too would cause the key switch to be unpowered.
Go back and check the configuration of the battery positive cable at the MTD solenoid. If the small wire is already molded into the cable end eye terminal, no further changes need to be made on that, but the fuse might be blown. Note that sometimes the small red wire is molded into the battery positive cable where the cable connects to the battery instead of at the solenoid post. If yours is like that, no further changes needed and check the fuse. I am going ahead with a test procedure tailored to your project. I was pushed for time really bad earlier and I put the above test in as Copy & Paste from a similar issue on a different thread (but sorta similar).
Check the points I listed here and post back please. Thank you for your patience.

It's 11:34 PM (Eastern Time) and I finally got the Briggs engine to crank because of your assistance... Thank you Mownie!!

I overlook something that was very basic. As stated on your last post, I connected the "battery positive" cable which had an attached small red wire from the solenoid to the starter and not from the "battery positive" to the solenoid. I connected another positive wire from the solenoid to the starter and the engine cranked!

Mownie, I appreciate your quick response... the reason it took me so long to get back was I had to change a solenoid/starter on my 98 minivan few days ago.

Tomorrow, I am going to work on the throttle and choke cable and the driving belt.

Here are few of my concerns/questions:
1. I tightened the electric clutch screw - turned it one revolution... would that cause any issues?
2. In you opinion, how long do you think my Briggs engine (ELS/XRD) will last? I had Kohler (Command Pro 25HP) for only 7 years and I kept it well maintained.
3. Should I use synthetic oil like Mobile 5W-30 or should I used conventional?
4. I have a steep incline in my back yard and I was wondering whether that caused the engine to go bad prematurely.
5. Because it's a new engine Throttle and Choke does not move all the way. Is this normal? Is there anyway that I can adjust this?

In summary, to change the engine on Craftsman GT500 from 25HP Kohler Engine (917.276020) to 27 Briggs ELS 44Q777:
1. Remove the Kohler Engine and disconnect harness from the engine.
2. Thread the 27 Briggs Engine to mount onto the tractor.
3. Buy a solenoid (~$10) and mount it on the tractor next to the battery since Kohler engine have a solenoid built into the engine. I bought one which had two posts and one connector.
4. Connect the battery positive cable which had an attached small red wire from the solenoid to the starter and not from the "battery positive" to the solenoid. Connect another wire from the solenoid to the starter.
5. Remove the white wire coming from the brake switch located near the battery. Unplug the cable end which goes to the the harness and plug it into the solenoid connector.
6. Connect back the harness. No rewiring necessary, except to unplug the white wire coming from the brake clutch and connect it to the solenoid.
7. Connect choke/throttle cables.

Once I get everything assembled, I am going to come back and let you know.

*************CORRECTION**********************

4. Connect the battery positive cable which had an attached small red wire from the "battery positive" to the solenoid. Connect another wire from the solenoid to the starter.

1. "I tightened the electric clutch screw - turned it one revolution... would that cause any issues?"
Ans. I don't know what screw that would be.??? I have a manual for testing and adjusting PTO clutches. I can e-mail a copy to you if you send me an e-mail.
2. "In you opinion, how long do you think my Briggs engine (ELS/XRD) will last?" Obviously, that is not an easy question to give a rigid answer to, too many variables. Ans. You can reasonably expect to get 500 plus hours out of the engine if you maintain it properly. This includes oil and filter change annually if you run it less than 50 hours per year. If you run more than 50 hours annually, change the oil and filter every 50 hours (or sooner if desired). Always change the oil at or near the last use of the season if your locale has an off season in winter. Do not store the machine with old, dirty oil in the engine.
The list of precautions and maintenance could go on for as long as we could discuss it on the forum.
ALWAYS check the engine oil BEFORE you use the machine, this means every time you put fuel in the tank by my definition. You get in the habit of doing that and you'll never have a low oil episode sneak up on you.
Check and adjust the engine valves annually. Briggs version of compression release absolutely requires that the valve clearance be correct to assure easy, dependable starting.
Use Marvel Mystery Oil (or a comparable quality "upper cylinder lubricant" in your fuel supply. The MMM will help prevent engine valves from sticking during storage and helps lube the fuel system parts and affords some protection from corrosion of metal parts in the fuel system. Use fuel stabilizer in your fuel too.
3."3. Should I use synthetic oil like Mobile 5W-30 or should I used conventional?
Ans. Synthetic offers some advantages in reduction of heat and might even tip the scales in your favor if you ever get into a situation of extreme operating conditions.
But the main thing to consider is use a high quality oil and change it often. The most expensive synthetic will not save your engine if you never change your oil. As to the weight of oil, follow the Briggs guidelines in their operator manual.
4. "I have a steep incline in my back yard and I was wondering whether that caused the engine to go bad prematurely."
Ans. I don't know if that could be a factor or not. If you are scared when you mow it, and you ever ran it low of oil when on the slope, you could have contributed to the engine failure. But if a mouse had built a nest in the engine cooling shroud and caused the cylinder head to overheat and seize the valve, which the piston struck etc., that would have contributed too, but I just don't know.
5. "Because it's a new engine Throttle and Choke does not move all the way. Is this normal? Is there anyway that I can adjust this?"
Ans. Other than to make certain that you set the cable anchor screw and holding bracket so you actually get the full range of movement, there's not a lot I can tell you. Heck, the two manuals we have been referrencing show the throttle cables to have different part numbers, but that does not prove that one of them affords more range of movement. It could only mean a longer cable.

Mownie, thank you for the recommendations... I didn't even think about using Marvel Mystery Oil, but from now on I am going to use Marvel Mystery Oil. I just started using Seafoam not too long ago. Initially I poured in 5W-30 Mobile synthetic and I got lots of smoke from the engine. So I completely drained the oil today, and put in a conventional 5W-30 oil and no smoke.

I still need to replace it with a longer choke cable because it's too short, but my garden tractor is working. I mowed my lawn for the first time in 2 weeks, but after reading this forum, I just hope that the new Briggs Engine lasts. As I mowed today, I found that Briggs engine were little lounder than my previous Kohler. However, my 27 HP Briggs engine had noticeably more mowing power than the 25 HP Kohler.

Thank you Mownie for your assistance!

You are welcome, and be careful on that hill. Good luck.