find the remainder when 1021^1022 is divided by 1023?
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Step-by-step explanation:
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Answer:
《 1,622 》
Using mod 10,001 ,
10k ≡ -1
1m ≡ -100
Let N=1,001,001²
①N²
= (1m+1k+1)²
={ [-100]+1,001 }²
= 901²
= (900+1)²
=810k+1,801
≡ [-81]+1,801
≡1,720
②N^4
= 1720²
≡ (1700+20)²
≡ 2m+890k+68k+400
≡ [-200–89–6]+8k+400
≡ 400–295+8k
≡ 8105
③ N^8
≡ 8105²
≡ (8100+5)²
≡ (65m+610k)+81k+25
≡ [-6,500–61–7]+11,025
≡ 4,457
④N^16
≡ 4457²
≡ (4500–43)²
≡ (20m+250k)-387k+1849
≡ [-2k-25+39]+4849
≡ 2,863
⑤ N^32
≡ 2,863²
≡ (2800+63)²
≡ (7m+840k)+100(56)(63)+63²
≡ [-700–84]+(352k+800)+3,969
≡ -784+[-35]+2k+800+3,969
≡ 5,950
⑥N^64
≡ 5950²
≡ (6k–50)²
≡ 36m-600k+2500
≡ [-3600+60]+2500
≡ -1,040+[10k+1]
≡ 8,961
⑦N^128
≡ 8961²
≡ (9k–39)²
≡ 81m-702k+1521
≡ [-8,100+70]-479
≡ -8,509+[10k+1]
≡ 1,492
⑧ N^256
≡ 1492²
≡(1500–8)²
≡ (2m+250k)-24k+64
≡ [-200–25+2]-4k+64+[10k+1]
≡ 6,067–225
≡ 5,842
⑨ N^512
≡ 5,842²
≡ (6k-158)²
≡ 36m-1896k+(150+8)²
≡ [-3600+189]-6k+22k+500+2400+64
≡ 10k+6k-1200+689+64
≡ [-1]+4800+753
≡ 5,552
⑨ N^1024
≡ 5,552²
≡ (5,500+52)²
≡ 30m+250k+(52×11k)+2,704
≡ [-3k-25]+572k+2,704
≡ -3k–25+[-57]+4704
≡ 1,704–82
≡ 1,622
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Step-by-step explanation: