140 Followers Problem - 2

\[\Large \lfloor \log (x) \rfloor + \lfloor \log (y) \rfloor + \lfloor \log (z) \rfloor = 140 \]

If the number of positive integral solutions to above equation can be expressed in the form of \(a^{b}\times c^{d} \times e^{f} \times g^{h} \times i^{j}\), where \(a\), \(c\), \(e\), \(g\) and \(i\) are distinct positive primes and \(b\), \(d\), \(f\), \(h\) and \(j\) are positive integers. Then find the value of

\[ a+b+c+d +e+f+g+h+i+j \]

Details and Assumptions

  • \(\lfloor k \rfloor \) represents the greatest integer less than or equal to \(k\).

  • \(\log k\) is the logarithm of \(k\) to the base 10.

Try its sister problem 140 Followers Problem.
This is my original problem.
This problem is from the set 140 Followers Problems.

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