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Topic review (newest first)
- ganesh
- 2005-11-17 17:59:08
It may not be true that you are bad at Math, Alisha...the topic we're discussing is not taught at School...simple as that 
- Alisha
- 2005-11-17 03:10:59
I must be really bad at moth cause I have no clue what you guys are talking about!
- mathsyperson
- 2005-11-16 03:42:31
Woo! For the number that ends in 5, the partner to the one you found, if you apply my proof to that then you will eventually get that the last digit is the last digit of 10y+c, and so would also be the last digit of c. It's much easier to continue the chain of the number ending in 5 because of that property and, as mentioned earlier, the one ending in 6 adds to the one ending in 5 to give 1000...0001, so you can use that to work out the one ending in 6 much quicker.
- ganesh
- 2005-11-15 23:34:56
mathsyperson wrote:
I'm not good at proofs, but this is my attempt:
Take 109376 as an example.
We need to prove that (y109376)² ends in y109376 for only one value of y.
y109376 can be written as y*10^6+1*10^5+0*10^4+9*10^3+3*10^2+7*10^1+6*10^0.
We are only interested in the 10^6 coefficient of the square of this. That will be equal to 2(y*(6*10^0))+a constant made up of the expansion of the other powers of 10, but that does not involve y.
So, the 10^6 coefficient is 12y+c, which should only be equal to y for one value of y.
We are only interested in the last digit of 12y+c, so it can be called 2y+c without affecting our purposes. However big c is, it can also have all of its digits except the last removed.
We want to solve y=2y+c.
Take away y+c: y=-c
Give back one of c's 10's to make it more sensible: y=10-c
c is a constant, so there is only one value for y.
As shown in the expansion of y109376, the important coefficient will always take the form 2(y*6), because 6 is always the last digit. This means that this proof can be carried forward for any point on the chain of the magic number.[/proof]
There's probably a flaw in there somewhere, could someone check it please?
Thanks a million, Mathsyperson! I have checked the proof thoroughly and now, I can declare the proof is precise, elegant, and flawless.
This proof has been very helpful to me.
I thank you, again, MathsIsFun, for the Full Precision Calculator.
- mathsyperson
- 2005-09-28 23:47:39
How very stretchy of you.
- John E. Franklin
- 2005-08-30 01:40:56
Answer to original question at top was correct according to "Just BASIC" program I wrote:
2 4
3 8
4 16
5 32
6 64
7 128
8 256
9 512
10 1024
11 2048
12 4096
13 8192
14 16384
15 32768
16 65536
17 31072
18 62144
19 24288
20 48576
21 97152
22 94304
23 88608
24 77216
25 54432
26 8864
27 17728
28 35456
29 70912
30 41824
31 83648
32 67296
33 34592
34 69184
35 38368
36 76736
37 53472
38 6944
39 13888
40 27776
41 55552
42 11104
43 22208
44 44416
45 88832
46 77664
47 55328
48 10656
49 21312
50 42624
51 85248
52 70496
53 40992
54 81984
55 63968
56 27936
57 55872
58 11744
59 23488
60 46976
61 93952
62 87904
63 75808
64 51616
65 3232
66 6464
67 12928
68 25856
69 51712
70 3424
71 6848
72 13696
73 27392
74 54784
75 9568
76 19136
77 38272
78 76544
79 53088
80 6176
81 12352
82 24704
83 49408
84 98816
85 97632
86 95264
87 90528
88 81056
89 62112
90 24224
91 48448
92 96896
93 93792
94 87584
95 75168
96 50336
97 672
98 1344
99 2688
100 5376
101 10752
102 21504
103 43008
104 86016
105 72032
106 44064
107 88128
108 76256
109 52512
110 5024
111 10048
112 20096
113 40192
114 80384
115 60768
116 21536
117 43072
118 86144
119 72288
120 44576
121 89152
122 78304
123 56608
124 13216
125 26432
126 52864
127 5728
128 11456
129 22912
130 45824
131 91648
132 83296
133 66592
134 33184
135 66368
136 32736
137 65472
138 30944
139 61888
140 23776
141 47552
142 95104
143 90208
144 80416
145 60832
146 21664
147 43328
148 86656
149 73312
150 46624
151 93248
152 86496
153 72992
154 45984
155 91968
156 83936
157 67872
158 35744
159 71488
160 42976
161 85952
162 71904
163 43808
164 87616
165 75232
166 50464
167 928
168 1856
169 3712
170 7424
171 14848
172 29696
173 59392
174 18784
175 37568
176 75136
177 50272
178 544
179 1088
180 2176
181 4352
182 8704
183 17408
184 34816
185 69632
186 39264
187 78528
188 57056
189 14112
190 28224
191 56448
192 12896
193 25792
194 51584
195 3168
196 6336
197 12672
198 25344
199 50688
200 1376
201 2752
202 5504
203 11008
204 22016
205 44032
206 88064
207 76128
208 52256
209 4512
210 9024
211 18048
212 36096
213 72192
214 44384
215 88768
216 77536
217 55072
218 10144
219 20288
220 40576
221 81152
222 62304
223 24608
224 49216
225 98432
226 96864
227 93728
228 87456
229 74912
230 49824
231 99648
232 99296
233 98592
234 97184
235 94368
236 88736
237 77472
238 54944
239 9888
240 19776
241 39552
242 79104
243 58208
244 16416
245 32832
246 65664
247 31328
248 62656
249 25312
250 50624
251 1248
252 2496
253 4992
254 9984
255 19968
256 39936
257 79872
258 59744
259 19488
260 38976
261 77952
262 55904
263 11808
264 23616
265 47232
266 94464
267 88928
268 77856
269 55712
270 11424
271 22848
272 45696
273 91392
274 82784
275 65568
276 31136
277 62272
278 24544
279 49088
280 98176
281 96352
282 92704
283 85408
284 70816
285 41632
286 83264
287 66528
288 33056
289 66112
290 32224
291 64448
292 28896
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294 15584
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297 24672
298 49344
299 98688
300 97376
301 94752
302 89504
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304 58016
305 16032
306 32064
307 64128
308 28256
309 56512
310 13024
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312 52096
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314 8384
315 16768
316 33536
317 67072
318 34144
319 68288
320 36576
321 73152
322 46304
323 92608
324 85216
325 70432
326 40864
327 81728
328 63456
329 26912
330 53824
331 7648
332 15296
333 30592
334 61184
335 22368
336 44736
337 89472
338 78944
339 57888
340 15776
341 31552
342 63104
343 26208
344 52416
345 4832
346 9664
347 19328
348 38656
349 77312
350 54624
351 9248
352 18496
353 36992
354 73984
355 47968
356 95936
357 91872
358 83744
359 67488
360 34976
361 69952
362 39904
363 79808
364 59616
365 19232
366 38464
367 76928
368 53856
369 7712
370 15424
371 30848
372 61696
373 23392
374 46784
375 93568
376 87136
377 74272
378 48544
379 97088
380 94176
381 88352
382 76704
383 53408
384 6816
385 13632
386 27264
387 54528
388 9056
389 18112
390 36224
391 72448
392 44896
393 89792
394 79584
395 59168
396 18336
397 36672
398 73344
399 46688
400 93376
401 86752
402 73504
403 47008
404 94016
405 88032
406 76064
407 52128
408 4256
409 8512
410 17024
411 34048
412 68096
413 36192
414 72384
415 44768
416 89536
417 79072
418 58144
419 16288
420 32576
421 65152
422 30304
423 60608
424 21216
425 42432
426 84864
427 69728
428 39456
429 78912
430 57824
431 15648
432 31296
433 62592
434 25184
435 50368
436 736
437 1472
438 2944
439 5888
440 11776
441 23552
442 47104
443 94208
444 88416
445 76832
446 53664
447 7328
448 14656
449 29312
450 58624
451 17248
452 34496
453 68992
454 37984
455 75968
456 51936
457 3872
458 7744
459 15488
460 30976
461 61952
462 23904
463 47808
464 95616
465 91232
466 82464
467 64928
468 29856
469 59712
470 19424
471 38848
472 77696
473 55392
474 10784
475 21568
476 43136
477 86272
478 72544
479 45088
480 90176
481 80352
482 60704
483 21408
484 42816
485 85632
486 71264
487 42528
488 85056
489 70112
490 40224
491 80448
492 60896
493 21792
494 43584
495 87168
496 74336
497 48672
498 97344
499 94688
500 89376
501 78752
502 57504
503 15008
504 30016
505 60032
506 20064
507 40128
508 80256
509 60512
510 21024
511 42048
512 84096
513 68192
514 36384
515 72768
516 45536
517 91072
518 82144
519 64288
520 28576
521 57152
522 14304
523 28608
524 57216
525 14432
526 28864
527 57728
528 15456
529 30912
530 61824
531 23648
532 47296
533 94592
534 89184
535 78368
536 56736
537 13472
538 26944
539 53888
540 7776
541 15552
542 31104
543 62208
544 24416
545 48832
546 97664
547 95328
548 90656
549 81312
550 62624
551 25248
552 50496
553 992
554 1984
555 3968
556 7936
557 15872
558 31744
559 63488
560 26976
561 53952
562 7904
563 15808
564 31616
565 63232
566 26464
567 52928
568 5856
569 11712
570 23424
571 46848
572 93696
573 87392
574 74784
575 49568
576 99136
577 98272
578 96544
579 93088
580 86176
581 72352
582 44704
583 89408
584 78816
585 57632
586 15264
587 30528
588 61056
589 22112
590 44224
591 88448
592 76896
593 53792
594 7584
595 15168
596 30336
597 60672
598 21344
599 42688
600 85376
601 70752
602 41504
603 83008
604 66016
605 32032
606 64064
607 28128
608 56256
609 12512
610 25024
611 50048
612 96
613 192
614 384
615 768
616 1536
617 3072
618 6144
619 12288
620 24576
621 49152
622 98304
623 96608
624 93216
625 86432
626 72864
627 45728
628 91456
629 82912
630 65824
631 31648
632 63296
633 26592
634 53184
635 6368
636 12736
637 25472
638 50944
639 1888
640 3776
641 7552
642 15104
643 30208
644 60416
645 20832
646 41664
647 83328
648 66656
649 33312
650 66624
651 33248
652 66496
653 32992
654 65984
655 31968
656 63936
657 27872
658 55744
659 11488
660 22976
661 45952
662 91904
663 83808
664 67616
665 35232
666 70464
667 40928
668 81856
669 63712
670 27424
671 54848
672 9696
673 19392
674 38784
675 77568
676 55136
677 10272
678 20544
679 41088
680 82176
681 64352
682 28704
683 57408
684 14816
685 29632
686 59264
687 18528
688 37056
689 74112
690 48224
691 96448
692 92896
693 85792
694 71584
695 43168
696 86336
697 72672
698 45344
699 90688
700 81376
701 62752
702 25504
703 51008
704 2016
705 4032
706 8064
707 16128
708 32256
709 64512
710 29024
711 58048
712 16096
713 32192
714 64384
715 28768
716 57536
717 15072
718 30144
719 60288
720 20576
721 41152
722 82304
723 64608
724 29216
725 58432
726 16864
727 33728
728 67456
729 34912
730 69824
731 39648
732 79296
733 58592
734 17184
735 34368
736 68736
737 37472
738 74944
739 49888
740 99776
741 99552
742 99104
743 98208
744 96416
745 92832
746 85664
747 71328
748 42656
749 85312
750 70624
751 41248
752 82496
753 64992
754 29984
755 59968
756 19936
757 39872
758 79744
759 59488
760 18976
761 37952
762 75904
763 51808
764 3616
765 7232
766 14464
767 28928
768 57856
769 15712
770 31424
771 62848
772 25696
773 51392
774 2784
775 5568
776 11136
777 22272
778 44544
779 89088
780 78176
781 56352
782 12704
783 25408
784 50816
785 1632
786 3264
787 6528
788 13056
789 26112
790 52224
791 4448
792 8896
793 17792
794 35584
795 71168
796 42336
797 84672
798 69344
799 38688
800 77376
801 54752
802 9504
803 19008
804 38016
805 76032
806 52064
807 4128
808 8256
809 16512
810 33024
811 66048
812 32096
813 64192
814 28384
815 56768
816 13536
817 27072
818 54144
819 8288
820 16576
821 33152
822 66304
823 32608
824 65216
825 30432
826 60864
827 21728
828 43456
829 86912
830 73824
831 47648
832 95296
833 90592
834 81184
835 62368
836 24736
837 49472
838 98944
839 97888
840 95776
841 91552
842 83104
843 66208
844 32416
845 64832
846 29664
847 59328
848 18656
849 37312
850 74624
851 49248
852 98496
853 96992
854 93984
855 87968
856 75936
857 51872
858 3744
859 7488
860 14976
861 29952
862 59904
863 19808
864 39616
865 79232
866 58464
867 16928
868 33856
869 67712
870 35424
871 70848
872 41696
873 83392
874 66784
875 33568
876 67136
877 34272
878 68544
879 37088
880 74176
881 48352
882 96704
883 93408
884 86816
885 73632
886 47264
887 94528
888 89056
889 78112
890 56224
891 12448
892 24896
893 49792
894 99584
895 99168
896 98336
897 96672
898 93344
899 86688
900 73376
901 46752
902 93504
903 87008
904 74016
905 48032
906 96064
907 92128
908 84256
909 68512
910 37024
911 74048
912 48096
913 96192
914 92384
915 84768
916 69536
917 39072
918 78144
919 56288
920 12576
921 25152
922 50304
923 608
924 1216
925 2432
926 4864
927 9728
928 19456
929 38912
930 77824
931 55648
932 11296
933 22592
934 45184
935 90368
936 80736
937 61472
938 22944
939 45888
940 91776
941 83552
942 67104
943 34208
944 68416
945 36832
946 73664
947 47328
948 94656
949 89312
950 78624
951 57248
952 14496
953 28992
954 57984
955 15968
956 31936
957 63872
958 27744
959 55488
960 10976
961 21952
962 43904
963 87808
964 75616
965 51232
966 2464
967 4928
968 9856
969 19712
970 39424
971 78848
972 57696
973 15392
974 30784
975 61568
976 23136
977 46272
978 92544
979 85088
980 70176
981 40352
982 80704
983 61408
984 22816
985 45632
986 91264
987 82528
988 65056
989 30112
990 60224
991 20448
992 40896
993 81792
994 63584
995 27168
996 54336
997 8672
998 17344
999 34688
1000 69376
1001 38752
1002 77504
1003 55008
1004 10016
1005 20032
1006 40064
1007 80128
1008 60256
1009 20512
1010 41024
1011 82048
1012 64096
1013 28192
1014 56384
1015 12768
1016 25536
1017 51072
1018 2144
1019 4288
1020 8576
1021 17152
1022 34304
1023 68608
1024 37216
1025 74432
1026 48864
1027 97728
1028 95456
1029 90912
1030 81824
1031 63648
1032 27296
1033 54592
1034 9184
1035 18368
1036 36736
1037 73472
1038 46944
1039 93888
1040 87776
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1042 51104
1043 2208
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1045 8832
1046 17664
1047 35328
1048 70656
1049 41312
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1051 65248
1052 30496
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1080 66176
1081 32352
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1086 35264
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1089 82112
1090 64224
1091 28448
1092 56896
1093 13792
1094 27584
1095 55168
1096 10336
1097 20672
1098 41344
1099 82688
1100 65376
1101 30752
1102 61504
1103 23008
1104 46016
1105 92032
1106 84064
1107 68128
1108 36256
1109 72512
1110 45024
1111 90048
1112 80096
1113 60192
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1115 40768
1116 81536
1117 63072
1118 26144
1119 52288
1120 4576
1121 9152
1122 18304
1123 36608
1124 73216
1125 46432
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1128 71456
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1133 86592
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1135 46368
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1147 23328
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1151 73248
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1160 2976
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1169 23712
1170 47424
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1172 89696
1173 79392
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1177 70272
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2500 9376
Sorry about the disk space I must be using somewhere...
- ganesh
- 2005-08-12 20:20:56
You are right, MathsisFun. I am working on this. I shall give you the details next week.
- MathsIsFun
- 2005-08-12 19:42:30
Is this something you are working on, ganesh? Something you want our help with?
- ganesh
- 2005-08-12 19:17:49
0^0^0 is not defined.
But for 1, as you pointed out, 2^Mx > x^x^x.
For this problem, let us take value of x ≥ 2, x ∈ N
- mathsyperson
- 2005-08-12 18:59:02
I hate to break it to you, but 2^Mx > x^x^x for x = 0 or 1.
- ganesh
- 2005-08-12 14:10:13
If Mx is the Least Common Multiple of all numbers from 1 to x,
can it be proved that
x^x < 2^Mx < x^x^x ?
The second part is very simple. 2^Mx is certainly less than x^x^x because both the base and the exponent of 2^Mx are lesser than x^x^x.
- ganesh
- 2005-07-15 15:57:42
ganesh wrote:
I discovered that 2^20 ends in 76.
Thereafter, any number of the form 2^20n ends in 76.
Similarly, any number of the form 2^100n ends in 376.
Next, I had to know the last few digits of 2^500 and 2^2500.
When I learnt that they are 9376 and 09376, I was excited.
Because, any number of the form 2^500n would then have to end in 9376
and every 2^2500n would have to end in 09376.
This is what Rora was talking about. You are too young to understand this, Maxine. This is about higher powers of the number 2.
- Maxine
- 2005-07-15 03:52:29
Roraborealis wrote:
Wow, you discovered a new number pattern thingy!
sorry because i dident understand ur story
Maxine
- MathsIsFun
- 2005-07-14 17:59:01
Ahh... so it was YOU that got me to admit to that LOL
- MathsIsFun
- 2005-07-14 17:58:02
It just struck me that every such proof usually starts with the 1, 2 example ... and I just assumed it was necessary.
But really if you prove that it moves from "n" to "n+1" that should be enough - it is proven all up and down the natural number scale, including 1.
(I think!)