All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Numbers
Year by Year (Posted on 2004-11-23) Difficulty: 4 of 5
Consider N=2004^2004.

1) What are the first 3 digits of N?

2) What are the last 3 digits of N?

See The Solution Submitted by SilverKnight    
Rating: 2.3333 (6 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re: alternate method (not tested) | Comment 3 of 8 |
(In reply to alternate method (not tested) by Larry)

Your alternate method for part 2 is basically the same as mine, except broken down into smaller steps--really small steps in this case.

The alternate method for part 1 won't work, as inaccuracies will creep up from lower order positions, similar to the way when memory sizes are expressed in K, the difference between a factor of 1000 and a factor of 1024 creeps up into the very first digits when you get into G's.

Here's the method carried out, with the method in the left side and the actual powers in the right (when large enough, expressed in scientific notation):

4016          4016016
8048          8048096064
1612          16128384512256
3230          3.232128256256102D+16
6472          6.477185025537229D+19
1296          1.298027879117661D+23
2597          2.601247869751792D+26
5204          5.212900730982591D+29
1042          1.044665306488911D+33
2088          2.093509274203778D+36
4184          4.195392585504371D+39
8384          8.40756674135076D+42
1680          1.684876374966692D+46
3366          3.376492255433251D+49
6745          6.766490479888235D+52
1351          1.356004692169602D+56
2707          2.717433403107883D+59
5424          5.445736539828198D+62
1086          1.091325602581571D+66
2176          2.187016507573468D+69
4360          4.38278108117723D+72
8737          8.783093286679169D+75
1750          1.760131894650506D+79
3507          3.527304316879613D+82
7028          7.068717851026745D+85
1408          1.41657105734576D+89
2821          2.838808398920903D+92
5653          5.688972031437488D+95
1132          1.140069995100073D+99
2268          2.284700270180545D+102
4545          4.578539341441813D+105
9108          9.175392840249393D+108
1825          1.838748725185978D+112
3657          3.684852445272701D+115
7328          7.384444300326492D+118
1468          1.479842637785429D+122
2941          2.965604646122D+125
5893          5.943071710828488D+128
1180          1.190991570850029D+132
2364          2.386747107983458D+135

Note that already by 2004^5 = 3.232128256256102D+16, the alternate method yields 3230 -- incorrect in the 4th digit.  The last line above represents 2004^41, and by then the 3rd digit is incorrect.

For 2004^2004, the alternate method produces 6757 as the last 4-digit number, indicating 675 as the first 3 digits, which is not correct.

The following table, at intervals of 50 in the power, show the proposed alternate method followed by the actual value expressed in scientific notation:

 power       alt. method     2004^power  
 50            1231         1.244188 x10^  165
 100           1518         1.548003 x10^  330
 150           1871         1.926006 x10^  495
 200           2308         2.396312 x10^  660
 250           2841         2.981462 x10^  825
 300           3494         3.709498 x10^  990
 350           4304         4.615312 x10^  1155
 400           5308         5.742313 x10^  1320
 450           6537         7.144515 x10^  1485
 500           8056         8.889116 x10^  1650
 550           9933         1.105973 x10^  1816
 600           1223         1.376038 x10^  1981
 650           1508         1.712049 x10^  2146
 700           1858         2.130110 x10^  2311
 750           2284         2.650256 x10^  2476
 800           2811         3.297416 x10^  2641
 850           3462         4.102604 x10^  2806
 900           4268         5.104409 x10^  2971
 950           5260         6.350842 x10^  3136
 1000          6480         7.901639 x10^  3301
 1050          7989         9.831121 x10^  3466
 1100          9837         1.223176 x10^  3632
 1150          1213         1.521860 x10^  3797
 1200          1494         1.893480 x10^  3962
 1250          1838         2.355844 x10^  4127
 1300          2260         2.931112 x10^  4292
 1350          2785         3.646853 x10^  4457
 1400          3434         4.537369 x10^  4622
 1450          4232         5.645338 x10^  4787
 1500          5212         7.023859 x10^  4952
 1550          6424         8.738998 x10^  5117
 1600          7913         1.087295 x10^  5283
 1650          9757         1.352799 x10^  5448
 1700          1201         1.683136 x10^  5613
 1750          1478         2.094137 x10^  5778
 1800          1818         2.605499 x10^  5943
 1850          2240         3.241730 x10^  6108
 1900          2761         4.033320 x10^  6273
 1950          3404         5.018207 x10^  6438
 2000          4192         6.243590 x10^  6603
 2004          6757         1.006990 x10^  6617

 


  Posted by Charlie on 2004-11-23 16:28:43
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (3)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information