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2^(10n) (Posted on 2015-04-15) Difficulty: 2 of 5
210 = 1024
220 = 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.

See The Solution Submitted by Jer    
Rating: 1.0000 (1 votes)

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Solution worked out | Comment 4 of 8 |
The mantissa of log(1024) is .010299956639812 (common logs), while that of 2 is  .3010299956639811; dividing the first into the second gives 29.22633620615765, so that 1024^29 will still begin with 1 but 1024^30 will begin with 2.

Checking via calculator app:

1024^29 ~=  1.989292945639148 * 10^87
1024^30 ~=  2.037035976334467 * 10^90

via UBASIC, these two numbers are:

1989292945639146568621528992587283360401824603189390869761855907572637988050133502132224
2037035976334486086268445688409378161051468393665936250636140449354381299763336706183397376
 
The powers of 2 of course are 10 times the powers of 1024: 290 and 300, the latter being the sought answer. 

  Posted by Charlie on 2015-04-15 09:01:19
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