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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

But try the problem first…

Answer: divisible by 37 and 101

Source

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.