Number Theory

If we wanted to fingd the last digit, we would have to go threw the endings of the powers of 7

 

7^1=7

7^2=9

7^3=3

7^5=1

7^6=7

 

So from here the cycle repeats every 4. 

2017 divided by 4 leaves a remainder of 1, so the last digit is 7. 

 

But the question asks for the last 10 and first 10 digits, and i have no idea how to solve that, I don't think that it is really posible, it would leave a huge number.

 

Sorry if this post was unhelpful, but I don't think that the solution is possible (I mean, it is possiblr, but, like I said befor, it would end up as a huge number that would be really hard to calculate, and I don't think there are any meathods or short cut). Once agian, sorry,

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This, according to calculator.net, is the answer.

 

Idk if it's correct but I set it to hgihest precision.

 

There must be an easier way. 

Unfortunately, most calculators on the Internet are faulty when handling very large numbers.

 

2017^(2017^2017). It is very easy to get the first 10 digits of big numbers such as this one:

 

1 - You must start from the RHS, what is inside the brackets: (2017^2017) =3.906 579 755 433 725 744 549 802 E+6665

2 - Take the log(base 10) of the first 2017 from the LHS and multiply by the number in (1) above:

Log(2017) x 3.906 579 755 433 725 744 549 802 E+6665 =1.291 009 715 962 041 616 156 219 E+6666. That is how this number begins.

 

3 - The last 10 digits are also very easy to get provided you have an accurate enough calculator that can handle very large numbers such as this one. The one "Big Calculator" is ,of course, WolframAlpha!. In this case, all you have to do is enter this into the "box" and you get the last 10 digits: 2017^(2017^2017) mod 10^10 and presto!: Last 10 digits =2,055,543,777