Happy Numbers Revisited
Here is how the "Happy Number" series is generated. Take any integer. Form the next term by adding together the sum of the squares of each of the term's digits. The succeeding terms are all generated in the same way. Continue until one of the terms is either 4 or 1. If it ends in 4, a continuous loop exists. If it ends in 1, the series terminates and the first term in such a series is known as a Happy Number.
Example: 49, 97, 130, 10, 1.
Find two different 3-digit happy numbers whose product is also a happy number, and which contains all the digits of the two 3-digit numbers once and once only.