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Divisibility and Integers | TOMATO B.Stat Objective 89

Try this TOMATO problem from I.S.I. B.Stat Entrance Objective Problem based on Divisibility and Integers. You may use sequential hints to solve the problem.

Try this problem from I.S.I. B.Stat Entrance Objective Problem based on Integers and divisibility.

Divisibility and Integers (B.Stat Objective Question )


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

Key Concepts


Integers

Remainders

Divisibility

Check the Answer


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

Try with Hints


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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