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It's okay to publish a book at any age. If it is a good book (or at least good enough.) Worse come to worse, if you can't publish at age 13, you can try again at age 18 or age 23. A famous writer, Pearl S. Buck published her first book at age 10, I believe.
Sending your book to a publisher requires more effort than just writing it. You also have to know how to submit your manuscript to a publisher. While the following steps won't guarantee your book will be published, they are good practices to follow when submitting your manuscript.