Problem of the Week Problem C Positively Perfect Prime Products

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Problem of the Week
Problem C
Positively Perfect Prime Products
A prime number is any number that has exactly two positive integer factors, 1
and the number itself. A composite number has more than two positive integer
factors. The number 1 is neither prime nor composite.
Four distinct prime numbers have a product of d10: a three-digit number with
hundreds digit d. Determine all possible values of the sum of these four prime
numbers.