How is the ISBN-13 check digit calculated?

Multiply the first 12 digits alternately by 1 and 3, add them up, and take (10 - sum mod 10) mod 10. For 978-0-306-40615 the sum is 93, so the check digit is 7.

Can every ISBN-13 be converted to ISBN-10?

Only ISBNs starting with 978. ISBNs starting with 979 have no ISBN-10 form.

Do I need a barcode on my KDP cover?

No. KDP can add one for you in the lower right of the back cover. If you supply your own, keep it in the barcode area of KDP's cover template.

Does a valid check digit mean KDP will accept my ISBN?

Not by itself. KDP also checks that the title, author and imprint match the ISBN agency record exactly.

ISBN validator, converter and barcode

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An ISBN's last digit is a check digit. For ISBN-13, multiply the first 12 digits alternately by 1 and 3, add them up, and the check digit is (10 - sum mod 10) mod 10. Paste an ISBN-10 or ISBN-13 to check it, convert between the two and download an EAN-13 barcode as SVG.

Check digit: -
ISBN-13: -
ISBN-10: -

The check digit formulas

ISBN-13: sum of the first 12 digits weighted 1, 3, 1, 3 and so on; check = (10 - sum mod 10) mod 10.
ISBN-10: sum of the first 9 digits weighted 10 down to 2; check = (11 - sum mod 11) mod 11, written as X when it is 10.
ISBN-10 to 13: prefix 978, drop the old check digit and work out a new ISBN-13 check digit.

Example: 978-0-306-40615-7 gives a weighted sum of 93 for its first 12 digits, so the check digit is 7 and the ISBN is valid. Its ISBN-10 is 0306406152.

The barcode is a standard EAN-13 drawn in your browser. Print it at 100% and check that it scans before relying on it.

Questions

How is the ISBN-13 check digit calculated?: Multiply the first 12 digits alternately by 1 and 3, add them up, and take (10 - sum mod 10) mod 10. For 978-0-306-40615 the sum is 93, so the check digit is 7.
Can every ISBN-13 be converted to ISBN-10?: Only ISBNs starting with 978. ISBNs starting with 979 have no ISBN-10 form.
Do I need a barcode on my KDP cover?: No. KDP can add one for you in the lower right of the back cover. If you supply your own, keep it in the barcode area of KDP's cover template.
Does a valid check digit mean KDP will accept my ISBN?: Not by itself. KDP also checks that the title, author and imprint match the ISBN agency record exactly.

The rules behind it

Sources

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