Question 1131959
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There are lots of things we can do using estimation and logical reasoning to determine that the integer is 2.<br>
To begin with, the units digit is 2, so the integer has to be even.  After that....<br>
(1)10^41 is a number with 42 digits.  If the 41st power of an integer is a number with only 13 digits, then the integer has to be VERY small.  So just with that much reasoning, the integer almost has to be 2.<br>
(2) We can gain confidence that the integer is 2 by noting that the units digit of the given number is 2.  No power of either 4 or 6 has units digit 2, so the only possibilities for the integer are 2 and 8.  And 8 is almost certainly too big.<br>
(3) Finally, to confirm that 2 is the integer we are looking for, we can use the (very useful!) approximation 2^10 = 10^3 to see that 2^41 will be a number with about 13 digits:<br>
2^41 = 2(2^40) = 2((2^10)^4) = 2((10^3)^4) = 2(10^12)<br>
That is a 13-digit number with first digit 2 -- which is what the given number is.