N=11^100+22^100+33^100+44^100+55^100+66^100+77^100+88^100+99^100
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No.103GAI2022幎6æ13æ¥ 10:20
以äžåååŒã¯ãã¹ãŠmod10
2^4=16â¡6
3^4=81â¡1
4^2=16â¡6
5^nâ¡5
6^nâ¡6
7^4=2401â¡1
8^4=4096â¡6
9^2=81â¡1
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Nâ¡1^100+2^100+3^100+4^100+5^100+6^100+7^100+8^100+9^100
=1+(2^4)^25+(3^4)^25+(4^2)^50+5^100+6^100+(7^4)^25+(8^4)^25+(9^2)^50
â¡1+6+1+6+5+6+1+6+1=33â¡3
ãªã®ã§ã3ã
No.104ãããã2022幎6æ13æ¥ 22:18
äžåŒâ¡1+0+1+0+1+0+1+0+1â¡1 (mod2)
ãŸããp=5 ã«ã€ããŠãã§ã«ããŒã®å°å®çãçšãããšã
äžåŒâ¡1+1+1+1+0+1+1+1+1â¡3 (mod5)
ãã£ãŠ
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No.105DD++2022幎6æ13æ¥ 22:29
ãäºäººãšãæ§è³ªãçç¥ãããŠããããšããã³ãã³äŒãã£ãŠããŸãã
ã²ãããªããšãã5ä¹ã«ãããŠã¯,a=1,2,3,,9ã§
a^5â¡a (mod 10)
ãæç«ããŠããããšã«æ°ä»ããŠããããæŽ»çšã§ããåé¡ãšããŠäœæããŠãããŸããã
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ãšãªã£ãå Žåã¯ã©ã®æ§ã«å¯ŸåŠã§ããã®ã§ããïŒ
No.106GAI2022幎6æ14æ¥ 05:36
> ãªããããmod 100ãšãªã£ãå Žåã¯ã©ã®æ§ã«å¯ŸåŠã§ããã®ã§ããïŒ
以äžåååŒã¯ãã¹ãŠmod100
n^2â¡nã®è§£ã¯0,1,25,76
(ã€ãŸããã®4ã€ã¯äœä¹ããŠãmod100ã§äžå€)
2^20=1048576â¡76
3^20=3486784401â¡1
4^10=1048576â¡76
5^2=25â¡25
6^5=7776â¡76
7^4=2401â¡1
8^20=1152921504606846976â¡76
9^10=3486784401â¡1
11^10=25937424601â¡1
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N=11^100+22^100+33^100+44^100+55^100+66^100+77^100+88^100+99^100
=11^100(1^100+2^100+3^100+4^100+5^100+6^100+7^100+8^100+9^100)
={(11^10)^10}{1+(2^20)^5+(3^20)^5+(4^10)^10+(5^2)^50+(6^5)^20+(7^4)^25+(8^20)^5+(9^10)^10}
â¡1ã»(1+76+1+76+25+76+1+76+1)
=333â¡33
ãšãªãã100ã§å²ã£ãäœãã¯33ã
No.107ãããã2022幎6æ14æ¥ 09:58
ãªã€ã©ãŒã®ããŒã·ã§ã³ã颿°ã䜿ããŸãã
ãªã€ã©ãŒã®å®çãããa ã 5 ã®åæ°ã§ãªããšã
a^Ï(25)=a^20â¡1 (mod25)
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äžåŒâ¡1+0+1+0+1+0+1+0+1â¡1 (mod4)
ãã£ãŠ
äžåŒâ¡33 (mod100)
No.108DD++2022幎6æ14æ¥ 11:16
ã€ãã§ã« 1000 ã§å²ãå Žåãã
ãªã€ã©ãŒã®å®çãããa ã 5 ã®åæ°ã§ãªããšã
a^Ï(125)=a^100â¡1 (mod125)
ãªã®ã§ã
äžåŒâ¡1+1+1+1+0+1+1+1+1â¡8 (mod125)
ãŸãã
äžåŒâ¡1+0+1+0+1+0+1+0+1â¡5 (mod8)
ãã£ãŠ
äžåŒâ¡133 (mod1000)
10000 ã§å²ããšãªããšæãå€ããªããšãããŸãããã
No.109DD++2022幎6æ14æ¥ 11:21
(303-n)/(320+n)=23/47
ã®è§£ã¯ 983 ã§ã¯ãªã 98.3 ã§ãããæŽæ°ã§ã¯ãªãã®ã§ã¯ã
No.99DD++2022幎6æ11æ¥ 12:52
303/320=3030/3200ããšããèŠæ¹ã¯ãéªéã§ãããïŒ
No.100HP管çè
2022幎6æ11æ¥ 17:56 å顿ã®è¡šçŸããååããåŒããã忝ã«è¶³ãããªããã§ãã®ã§ã
åé¡ïŒ303/320 ã®ååã¯äœã
è§£çïŒ3030
ãæ£è§£ãšããã¹ããã©ãããšãã話ã«ãªããŸããã
ç§ã¯äžæ£è§£ãšãã¹ãã ãšæããŸããã©ãã§ãããïŒ
No.101DD++2022幎6æ11æ¥ 19:37
ãŸãããã®è«ãèªããå Žåã303/320 ã 6060/6400 ãšã¿ãªãããšã§ 1966 ãªã©å¥ã®è§£ãèªããããŠããŸããŸããã
極端ã«ã¯ 303/320 ã (
303/98.3)/(320/98.3) ãšèŠåãã° 1 ãè§£ã«ãªãããã®åé¡ã®çãã¯ãä»»æã®æŽæ°ãã«ãªãããšæããŸãã
No.102DD++2022幎6æ12æ¥ 01:00
(5^2+8^2-7^2)/(2*5*8)=1/2ãšããèšç®ãã
3蟺ã5,7,8ã®äžè§åœ¢ã®5ãš8ã®èŸºã§æãŸããè§ã®è§åºŠã¯60°ãšãããã
(æ¢ç¥ãšããã°èšç®äžèŠ)
å³ã®åœ¢ã¯3蟺ã5,7,8ã®äžè§åœ¢ã®5ã®èŸºãš8ã®èŸºã®å€åŽã«
ããããæ£äžè§åœ¢ããã£ã€ãã圢ãªã®ã§ãAB=5+8=13ã
No.93ãããã2022幎6æ9æ¥ 13:04
å³ã§ã®
AD=a,BC=b,CD=c,AB=xãšãããŠ,ãã®4ã€ãæŽæ°ãšãªããçµåã調ã¹ãã
(a,b,c)=(1,4,7)->x=9
=(2,5,7)->x=10
=(3,6,7)->x=11
=(4,7,7)->x=12
=(5,8,7)->x=13
=(6,9,7)->x=14
=(7,10,7)->x=15
ãããã
äžè¬ã«(a,b,c)=(n,n+3,7)->x=n+8 (n=1,2,3,)
ãŸãã¯
(a,b,c)=(1,6,7)->x=9
=(2,7,7)->x=10
=(3,8,7)->x=11
=(4,9,7)->x=12
=(5,10,7)->x=13
=(6,11,7)->x=14
=(7,12,7)->x=15
ãããã
äžè¬ã«(a,b,c)=(n,n+5,7)->x=n+8 (n=1,2,3,)
c=7ãšèšå®ããŠããããšããã€ã³ãã«ãªãããã§ãã
No.94GAI2022幎6æ9æ¥ 16:21
c=13 ã c=19 ã§ãå¯èœãªã®ã§ã¯ã
ãããã 6n+1 åçŽ å æ°ãå°ãªããšã 1 ã€æã£ãŠããããšãæ¡ä»¶ãããªãã§ããããã
No.95DD++2022幎6æ10æ¥ 11:41
確ãã«6*n+1åã®æ°ã¯
7^2=3^2+8^2-3*8
=5^2+8^2-5*8
13^2=7^2+15^2-7*15 =>(a,b,c)=(n,n+8,13)->x=n+15 (n=1,2,3,)
=8^2+15^2-8*15 =>(a,b,c)=(n,n+7,13)->x=n+15 (n,1,2,3,)ãæ§æã§ããã
19^2= 5^2+21^2- 5*21 =>(a,b,c)=(n,n+16,19)->x=n+21 (n=1,2,3,)
=16^2+21^2-16*21 =>(a,b,c)=(n,n+5,19)->x=n+21 (n=1,2,3,)
25^2=25^2+25^2-25*25(ããã¯äŸå€ïŒ
31^2=11^2+35^2-11*35=>(a,b,c)=(n,n+24,31)->x=n+35 (n=1,2,3,)
=24^2+35^2-24*35=>(a,b,c)=(n,n+11,31)->x=n+35 (n=1,2,3,)
以äžåæ§
37^2= 7^2+40^2-7*40
=33^2+40^2-24*40
43^2=13^2+48^2-13*48
=35^2+48^2-35*48
49^2=16^2+55^2-16*55
=39^2+55^2-39*55

ãš60°ã®è§åºŠãæããäžè§åœ¢ã®äžèŸºãäžããŠããïŒçµã
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No.96GAI2022幎6æ11æ¥ 07:47
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ããŠæå䞊ãã§ãã人æ°ã100,1000,10000(人)ã§ããå Žå
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No.90GAI2022幎6æ7æ¥ 07:35
çè«çã«èšç®ããæ¹æ³ã¯ããããŸããã§ããã®ã§ã
ããã°ã©ã ãäœã£ãŠèª¿ã¹ãŸããããã®çµæã¯
人æ°100,1000,10000,100000,1000000,10000000,100000000人ã«å¯ŸããŠ
(s,w)=(24, 97),(142, 997),(1015, 9997),(7986, 99997),
(66164, 999991),(565513, 9999985),(4944199, 99999967)
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No.91ãããã2022幎6æ7æ¥ 13:05
管ç人ããããã®ã³ã¡ã³ãã§
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No.92GAI2022幎6æ8æ¥ 09:00
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No.83GAI2022幎6æ3æ¥ 08:08
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44åã®äžã«å¶æ°ã®æ°ã ãã®digitsã§æ§æããããã®ã¯ãããŸããïŒ
No.85GAI2022幎6æ3æ¥ 11:17
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No.81GAI2022幎6æ2æ¥ 06:41
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2æ¡: æå€§2å (13,17,37,79)
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4æ¡: æå€§11å (1237,1279)
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6æ¡: æå€§148å (123479)
7æ¡: æå€§731å (1235789)
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9æ¡: æå€§26519å (102345697)
10æ¡: æå€§152526å (1123465789)
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No.79GAI2022幎5æ31æ¥ 17:57
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No.77HP管çè
2022幎5æ23æ¥ 17:50 A1,A2,A3,A4,A5,ãš
a1,a2,a3,a4,a5,ã®äžæè°ãªé¢ä¿ã§
A1=a1^2-a2^2
A2=2*a1*a2
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A1^2+A2^2=(a1^2-a2^2)^2+(2*a1*a2)^2
=a1^4-2*a1^2*a2^2+a2^4+4*a1^2*a2^2
=a1^4+2*a1^2*a2^2+a2^4
=(a1^2+a2^2)^2
A1=a1^2+a2^2-a3^2
A2=2*a1*a3
A3=2*a2*a3
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A1^2+A2^2+A3^2=(a1^2+a2^2-a3^2)^2+(2*a1*a3)^2+(2*a2*a3)^2
=(a1^2+a2^2)^2-2*(a1^2+a2^2)*a3^2+a3^4+4*(a1^2+a2^2)*a3^2
=(a1^2+a2^2)^2+2*(a1^2+a2^2)*a3^2+a3^4
=(a1^2+a2^2+a3^2)^2
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A1=a1^2+a2^2+a3^2-a4^2
A2=2*a1*a4
A3=2*a2*a4
A4=2*a3*a4
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A1^2+A2^2+A3^2+A4^2=(a1^2+a2^2+a3^2-a4^2)^2+(2*a1*a4)^2+(2*a2*a4)^2+(2*a3*a4)^2
=(a1^2+a2^2+a3^2)^2-2*(a1^2+a2^2+a3^2)*a4^2+a4^4+4*(a1^2+a2^2+a3^2)*a4^2
=(a1^2+a2^2+a3^2)^2+2*(a1^2+a2^2+a3^2)*a4^2+a4^4
=(a1^2+a2^2+a3^2+a4^2)^2
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A1=a1^2+a2^2+a3^2+a4^2++(an-1)^2-an^2
A2=2*a1*an
A3=2*a2*an
A4=2*a3*an
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An=2*(an-1)*an
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A1^2+A2^2+A3^2++An^2=(a1^2+a2^2+a3^2++an^2)^2
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No.76GAI2022幎5æ21æ¥ 09:17