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

The integer in question is 89.<o:p></o:p>

When the puzzle was on the board,  I have strongly objected to the d4 degree of difficulty.<o:p></o:p>

For high school graduate it is IMHO D2;
  for someone used to swim in the ocean of numbers it is  d1 or less .<o:p></o:p>

ITo get the correct answer it took 90 sec , most of the time (believe me) was spent to check the raising to the 13th power.<o:p></o:p>

I feel that I should explain , how the short mental process worked:l .<o:p></o:p>

a.        The integer n is a 2-digit number , much closer to 100 than
to 10. (There are 26 digits in the long number , and 100^13=10^26 i.e. the smallest 27 digit number).<o:p></o:p>

b.     This integer's last digit must be 9, since
13mod4= 1 and every integer's 4k+1)th power has the same last digit as the 1st power e.g   2,32,512,8192…3,243.19683   etc<o:p></o:p>

c.      99 was too close to  100, so my guess was  89.<o:p></o:p>

The facts in a. & b. are widely known . As to the last statement , one might start with 99 and be right on his 2nd guess.
 I  started with 89, estimating that 99^13=(100-1)^13 =
=(100^13)*(1-.01)^13 which can be approximated
 (100^13)(1-.13)=8.7*10^26. – much higher than 2.19<o:p>*10^26.</o:p>

<o:p>dixi.</o:p>

<o:p> </o:p>

Posted by Ady TZIDON on 2013-01-12 11:56:57