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I deposited dogecoins in my cryptsy account through faucet. How the transaction id (txid) has been displayed in cryptsy UI? suppose I make the transaction then I know the transaction id, but how cryptsy get transaction id for the particular address?

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Coins are associated with an address, but they are actually stored in transaction outputs.

Transactions take one (or several) inputs (referencing previous transactions) and sign over the total amount of the inputs to one (or several) outputs.

Usually, these outputs are formed in a way that they can be spend by the owner of a specific address, but other requirements to the signature can be scripted. Anyway, each and every balance is at atomic level made up from these transaction outputs.

The transaction outputs, though, reference the transaction that created them. That means, anyone looking at the blockchain can easily see what transaction sent the money to your address.

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