If sum of the number 985 and two other numbers obtained by arranging the digits of 985 in cyclic order is divided by 111, 22 and 37 respectively. Find the quotient in each case.

Given:

The sum of the number 985 and two other numbers obtained by arranging the digits of 985 in cyclic order is divided by 111, 22 and 37 respectively.

To do:

We have to find the quotient in each case.

Solution:

The given number is 985

The other two numbers by arranging its digits in cyclic order are $859, 598$

These numbers are of the form $\overline{abc},\overline{bca},\overline{cba}$

Therefore,

If $985 + 859 + 598$ is divided by 111, then quotient is $a + b + c = 9 + 8 + 5 = 22$

If this sum is divided by 22, then the quotient is $111$.

If this sum is divided by 37, then the quotient is $3(a+b+c)=3(22)=66$.