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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

- divisible neither by 37 nor by 101
- divisible by both 37 and 101
- divisible by 37 and not by 101
- divisible by 101 and not by37

Integers

Remainders

Divisibility

But try the problem first...

Answer: divisible by 37 and 101

Source

Suggested Reading

B.Stat Objective Problem 89

Challenges and Thrills of Pre-College Mathematics by University Press

First hint

here we take 300 digit number all digit 1s

Second Hint

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

Final Step

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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