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Question Number 795 by malwaan last updated on 15/Mar/15
what is the last digit of  7^((7^((7....)) ))    the number of 7′s is 1001
whatisthelastdigitof7(7(7.))thenumberof7sis1001
Answered by prakash jain last updated on 15/Mar/15
7^1 ≡3 (mod 4), 7^2 ≡1 (mod 4)  7^7 ≡3^4 3^3 ≡27≡3 (mod 4)  7^7 ≡3^(4k+3) ≡(3^4 )^k 3^3  ≡ 3 (mod 4)  7^7^(7(1000 times))  ≡3(mod 4)  7^(4k+3) =7^(4k) 7^3   7^(4k)  has last digit 1.  7^3  has last digits 3.  7^(4k+3)  will have last digit 3.
713(mod4),721(mod4)773433273(mod4)7734k+3(34)k333(mod4)777(1000times)3(mod4)74k+3=74k7374khaslastdigit1.73haslastdigits3.74k+3willhavelastdigit3.

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