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What is the sum of the two smallest distinct prime factors of 2^{27} + 3^{27}?'
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Answer. This sum is 5 + 7 = 12.
Solution
1. It is clear that 2 is not a factor of .
2. It is clear that 3 is not a factor of .
3. has the factor (2 + 3) = 5 since has this well known decomposition in the product of these factors:
=
Thus the first smallest prime factor of the sum is 5.
4. has the factor = 35 since can be factored in this way:
= = =
and (8+27) = 35 is multiple of 7, so the second smallest prime divisor of this sum is 7 (next after 5).
5. Therefore, 5 + 7 = 12 is the answer.
