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 Year by Year (Posted on 2004-11-23)
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)

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 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          40160168048          80480960641612          161283845122563230          3.232128256256102D+166472          6.477185025537229D+191296          1.298027879117661D+232597          2.601247869751792D+265204          5.212900730982591D+291042          1.044665306488911D+332088          2.093509274203778D+364184          4.195392585504371D+398384          8.40756674135076D+421680          1.684876374966692D+463366          3.376492255433251D+496745          6.766490479888235D+521351          1.356004692169602D+562707          2.717433403107883D+595424          5.445736539828198D+621086          1.091325602581571D+662176          2.187016507573468D+694360          4.38278108117723D+728737          8.783093286679169D+751750          1.760131894650506D+793507          3.527304316879613D+827028          7.068717851026745D+851408          1.41657105734576D+892821          2.838808398920903D+925653          5.688972031437488D+951132          1.140069995100073D+992268          2.284700270180545D+1024545          4.578539341441813D+1059108          9.175392840249393D+1081825          1.838748725185978D+1123657          3.684852445272701D+1157328          7.384444300326492D+1181468          1.479842637785429D+1222941          2.965604646122D+1255893          5.943071710828488D+1281180          1.190991570850029D+1322364          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

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