Brainsteem Mathematics Challenges: Integer Powers
Question
Which of the following is the highest number?
29, 38, 47, 56, 65, 74, 83, 92.
A reminder that these questions should be done without any electronic aid.
In this case, it is simple enough to just calculate each power, but that is time-consuming. Is there a way to speed up reaching the solution?
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Enjoy!
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47 = 16384 is your answer. In my case it helps that I simply memorized powers up to a certain point, including the powers of 2 up through 1048576 (220).
For those who don't want to brute-force the answer either, what they can do is try to turn everything into prime factorizations. So for example, 47 = (22)^7 = 214 and 83 = (23)^3 = 29.
65 becomes 25 x 35, and from there it's a matter of comparing 25 (32) against 33 (27) to see that 65 > 38. However, 29 (512) > 35 (243), so 47 is still greater.
You're eventually left with comparing 47 against 56 and 74, and only needing to calculate those. That's 16384 vs. 15625 vs. 2401.
You memorised them! lol. Actually, doesn't surprise me - if you try the follow-up question, there is also a tricky step that those who work a lot with numbers will know what to do.
I dislike books that call them "tricks"; they are not tricks, just knowledge of how numbers work!
If this was too easy, a harder, more general, version has been posted here: Brainsteem Mathematics Challenges: Integer Powers Function.
There is a little mathematical lemma that comes in very handy for this question. Does anybody know what it is?
Hint: it will reduce that list of numbers by half!
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