what-is-the-last-digit-of-7-7-7-the-number-of-7-s-is-1001-

Question Number 795 by malwaan last updated on 15/Mar/15

w h a t i s t h e l a s t d i g i t o f

7^{(7^{(7 \dots .)})}

t h e n u m b e r o f 7^{'} s i s 1001

Answered by prakash jain last updated on 15/Mar/15

7^{1} \equiv 3 (\mod 4), 7^{2} \equiv 1 (\mod 4)

7^{7} \equiv 3^{4} 3^{3} \equiv 27 \equiv 3 (\mod 4)

7^{7} \equiv 3^{4 k + 3} \equiv {(3^{4})}^{k} 3^{3} \equiv 3 (\mod 4)

7^{7^{7 (1000 times)}} \equiv 3 (\mod 4)

7^{4 k + 3} = 7^{4 k} 7^{3}

7^{4 k} has last digit 1 .

7^{3} has last digits 3 .

7^{4 k + 3} will have last digit 3 .