Prom numbers

A \(b\)-prom number is a positive integer which is the concatenation of the product of its digits and the sum of its digits in base \(b\). For example, \(3612\) is a 10-prom number, since the product of its digits is \(36\) and the sum of its digits is \(12\), and \(3612\) is the concatenation of \(36\) and \(12\). Note that \(1236\) is not a 10-prom number; the concatenation must be the product before the sum. Also note that \(01\) is not a 10-prom number; prom numbers may not begin with \(0\).

What is the only 3-digit 10-prom number?

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