The number 21982145917308330487013369 is the thirteenth power of a natural number. Find the number using just pen and paper.

Let the required thirteenth root be N, so that:

N^13 = 21982145917308330487013369

Firstly, since N^13 < 10^26, it follows that: N < 100

Secondly, for any positive integer k and for any positive integer n, we know that k^(4*n+1) and k share the same last digit. Substituting n=3, we observe that the last digit of N is 9.

So, N= 10s-1, for a positive integer s

Accordingly, (10s-1)^13 = 69 (mod 100)

or, 130s-1 = 69 (mod 100) (Expanding l.h.s by Binomial Theorem and reducing mod 100)

or, 30s = 70 (mod 100)

or, 3s=7 (mod 10)

or, s=9(mod 10)

Since N < 100, it follows that: s < 11, so that: s=9

Accordingly, N = 10*9-1=89

Consequently, the required thirteenth root is 89.

*Edited on ***January 12, 2013, 11:51 am**

*Edited on ***January 12, 2013, 12:05 pm**