2
10 = 1024
2
20 = 1048576
Note that raising 2 to each of the first two multiples of 10 results in a number whose first digit is 1.
Find the smallest multiple of 10 where 2 raising to that power results in a number that does not begin with 1.
(In reply to
re(3): worked out by Ady TZIDON)
I still do not understand your comment.
1024^1 = 2^10 begins with a 1; 1024^2 = 2^20 begins with a 1; etc.
A table of powers of 1024 = powers of 2^10 = powers of 2^(10k):
1 1024
2 1048576
3 1073741824
4 1099511627776
5 1125899906842625
6 1.152921504606845D+18 7 1.180591620717406D+21 8 1.20892581961463D+24
9 1.237940039285386D+27 10 1.267650600228232D+30
21 1.645504557321195D+63
22 1.684996666696936D+66
23 1.725436586697645D+69 24 1.766847064778371D+72 25 1.809251394333035D+75 26 1.85267342779701D+78 27 1.897137590064173D+81
28 1.942668892225694D+84 29 1.989292945639148D+87
At k=29, the powers of 2^(10k) still start with 1 and it is the last such.
At k = 30 this suddenly changes. Now they start with 2:
30 2.037035976334467D+90
31 2.085924839766534D+93 32 2.13598703592091D+96 33 2.18725072478299D+99 34 2.239744742177823D+102 35 2.29349861599007D+105 36 2.348542582773874D+108 37 2.404907604760423D+111 38 2.46262538727472D+114 39 2.521728396569217D+117
I don't disagree that there were numerous values that begin with 1; that was my point, and Jer's point in writing the puzzle.
BTW, you say " The 30 is the 1st number m, for which 2^(10m)begins with the digit 1." Actually it's the first that begins with 2, also my point.
What is the point of checking 2^291 through 2^299; those exponents are not multiples of 10. They are examples of numerous powers that both do and do not begin with 1. There are numerous examples of 2^k, lower than 2^300 that do not begin with 1, if that was your point, but none of those k were multiples of 10. The problem was really to find the power of 1024 that does not begin with 1, and that is what I was looking for, and that was the use of the word still.
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Posted by Charlie
on 2015-04-17 14:09:41 |
re(4): worked out
