Find the quotient and remainder when 300...007(number of 0s is 99) is divided by 37.
At the outset, we observe that:
(1000^r-1)/999
= (1000)^(r-1)+(1000)^(r-2)+.......+1000^2+1000+1
=1001001001......001 (2(r-1) zeros)
=> (1000^r-1)/37
= 27027027....027 (r-1 zeros)
=> 30(10^(3r)-1)/37 = 810810810......810810810 (r zeros) .....(i)
Let the given number be N
Then, N=3*(10^100)+7 = 30(10^99-1)+37
Accordingly,
N/37
= 30(10^99 -1)/37 +1
Substituting r=33 in (i), we have:
30(10^99-1)/37 = 810810810........810810810 (33 zeros)
Accordingly, N/37
= 810810810......810810810 (33 zeros) + 1
=810810810.....810810811 (32 zeros)
Consequently, the required quotient is 810810810.....810810811 (32 zeros) with NO remainder.
Edited on January 3, 2022, 9:27 am
Puzzle Solution
