Multiplication table puzzle

In the multiplication table on the right, only some of the numbers have been given. What is the value of A+B+C+D+E?
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Problem



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Multiplication Table Puzzle


In the multiplication table on the right, the row and column headings are all missing, and only some of the products in the table are filled in.

All the numbers in the table are positive integers.

What is the value of $A+B+C+D+E$?

If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.