Mistakes give rise to Problems- 13

Algebra Level 4

In maths, we do a×b=aba\times b=ab. But if you do that while there is the log function, log(a)×log(b)=log(a×b)\color{Red}{\log (a) \times \log (b) = \log (a\times b)} then that will be a big mistake!

But for some pairs of integers (a,b)(a,b), for which log(a)\log (a) and log(b)\log (b) are also integers, the above property is true. Find the sum of all aa and bb in these pairs.

If you get nn pairs (a1,b1),(a2,b2),,(an,bn)(a_1,b_1),(a_2,b_2),\ldots,(a_n,b_n), then answer should be reported as k=1n(ak+bk)\displaystyle \sum_{k=1}^n \biggl(a_k+b_k \biggr)

Details and assumptions:

  • Assume we take the log in base 10.

  • We only consider aa and bb as integers, and so are log(a)\log(a) and log(b)\log(b)

This problem is a part of the set Mistakes Give rise to problems.
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