Try this problem from I.S.I. B.Stat Entrance Objective Problem based on Integers and divisibility.
300 digit number with all digits equal to 1 is
Integers
Remainders
Divisibility
Answer: divisible by 37 and 101
B.Stat Objective Problem 89
Challenges and Thrills of Pre-College Mathematics by University Press
here we take 300 digit number all digit 1s
111...11=\(\frac{999...99}{9}\)(300 digits)
=\(\frac{10^{300}-1}{9}\)=\(\frac{(10^{3})^{100}-1}{9}\)=\(\frac{(10^{3}-1)X}{9}\)
since \(10^{3}-1\)=999 is divisible by 37 then 111...11(300 digits) is divisible by 37
111...11=\(\frac{999...99}{9}\)(300 digits)
=\(\frac{10^{300}-1}{9}\)=\(\frac{(10^{4})^{75}-1}{9}\)=\(\frac{(10^{4}-1)Y}{9}\)
since \(10^{4}-1\)=9999 is divisible by 101 then 111...11(300 digits) is divisible by 101.
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