Determine the quotient and remainder when 400....001 (100 zeros) is divided by 41.
Provide valid explanation for your answer.
The modular value of 40...01 repeats in a cycle of 5:
(showing number of zeros, followed by modular value)
[1, 32]
[2, 24]
[3, 26]
[4, 5]
[5, 0]
[6, 32]
[7, 24]
[8, 26]
[9, 5]
[10, 0]
[11, 32]
[12, 24]
[13, 26]
[14, 5]
[15, 0]
[16, 32]
[17, 24]
[18, 26]
[19, 5]
[20, 0]
from
clc
for lenz=1:20
s=['4' repmat('0',1,lenz) '1'];
n=str2sym(string(s));
disp([lenz mod(n,41)])
end
100 is divisible by 5 so the given number is divisible by 41, leaving zero remainder (which can be verified directly by extending the above program).
The quotient is
9756097560975609756097560975609756097560975609756097560975609756097560975609756097560975609756097561
That's 100 digits long, consisting of 19 repetitions of 97560 augmented by 97561 as the final five digits.
obtained by adding
disp(floor(n/41))
to the above program, and extending to having 100 zeros.
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Posted by Charlie
on 2022-08-27 09:22:34 |
computer solution (spoiler)
