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.
The supposed required number = 10*n (given)
Now, 2^(10*n)
= (1024)^n
= {(10)^(3n)} * (1.024)^n
Obviously, the smallest n occurs whenever:
(1.024)^n >= 2, since the other factor is only an exponent of 10.
or, n >= (log 2)/(log 1.024) ~= 29.226....(by calculator)
Since n > 29, it follows that the least positive integer value of n is 30.
This checks out by calculator, as:
(1.024)^30 ~= 2.037....
(1.024)^29 ~= 1.989 ....
Consequently, the required smallest multiple of 10 is 10*30 = 300
CHECK
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Calculator verification gives:
2^300 ~= 2.037036 * 10^90
As 2^300 begins with 2, it is verified that 300 is INDEED the sought for 10-multiple exponent of 2.
Edited on June 16, 2022, 12:45 am
Puzzle Solution: Easy Way
