A routing number beginning 10 confuses people because it looks like the start of a longer sequence rather than a code. It is a district number, and the district it names covers a large part of the middle of the country. That is the first thing this number tells you. The second is whether the digits hold together, which is settled by arithmetic and takes about ten seconds.
District ten, which is not a state
The opening pair is 10, the tenth Federal Reserve district, Kansas City. Districts are not states: the tenth covers a broad swathe of the centre and west of the country, and a single bank operating across it may use one routing number or several. So the prefix places the processing arrangement, not the branch, not the account and not the state. Anyone reading a state off the first two digits of a routing number is guessing.
Running the check digit
Weight the nine digits three, seven, one, repeating. For 103100195 that gives three times one plus one plus one, which is nine; seven times zero plus zero plus nine, which is sixty three; and three plus zero plus five, which is eight. Nine plus sixty three plus eight is eighty, which divides by ten. The number is well formed. Had a digit been transposed when it was copied to you, this test would almost certainly have failed, which is the whole point of the ninth digit.
Well formed against in use
The checksum tests internal consistency and nothing else. Roughly one in ten random nine digit strings passes it by chance, so a pass narrows very little. Whether the number is currently issued to a live institution is a question for the routing register, which is licensed and which this index does not republish. What this index does publish is what each bank states about its own identifiers on its own pages, dated, so that every fact here can be traced to its source.
The confirmation that matters
If you are about to send money, the useful step is not a lookup. Ask the payee which institution holds the account and confirm the routing number on that institution's own published page, then confirm the payee itself by telephone on a number you held beforehand. If the payment is a wire rather than an ACH entry, check specifically that you have the wire routing number, because many banks publish a different one and using the wrong one causes a return.
Questions people ask about 103100195 routing number
Is 103100195 a well formed routing number?
Yes. The weighted checksum comes to eighty, which divides by ten.
What does a prefix of 10 mean?
The tenth Federal Reserve district, Kansas City. It identifies a processing district, not a state or a branch.
Does passing the checksum mean the number is in use?
No. Around one in ten random nine digit strings passes by chance. The checksum catches typing errors, not invented numbers.
Is this the number to use for a wire?
Check with the institution. Many US banks publish separate routing numbers for ACH and for wire transfers.