Question

If the sum of the number 985 and the 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.
(a) 21, 111, 61
(b) 22, 111, 66
(c) 22, 111, 68
(d) None of these

Answer 1

Hint:To solve this question, we will first of all form two numbers using 985 and that too in cyclic order. After obtaining the two numbers, we will add them up to get their sum and then divide them one by one each by 111, 22, and 37 to get the quotient in each case.

Complete step by step answer:
The number is given as 985. We have to arrange 985 in the cyclic order to obtain two other numbers. The cyclic order is given as
\[9\to 8\to 5\]
Another number using 5 without changing the order can be
\[5\to 9\to 8\]
That is 598.
And the last number using 8 without changing the order can be
\[8\to 5\to 9\]
And that is 859.
So the three numbers possible are 485, 859 and 598.
Now, we will separately calculate the sum obtained when the sum of these are divided by 111, 22 and 37.
(1) When divided by 111
The sum of numbers 985, 859 and 598 is
\[\Rightarrow 985+859+598=2442\]
Dividing by 111, we have,
\[111\overset{22}{\overline{\left){\begin{align}
  & 2442 \\
 & \underline{222} \\
 & \text{ }222 \\
 & \underline{\text{ }222} \\
 & \text{ }000 \\
\end{align}}\right.}}\]
So the quotient is 22…..(i)
(2) When divided by 22
The sum is 2442. Divide by 22, we have,
\[22\overset{111}{\overline{\left){\begin{align}
  & 2442 \\
 & \underline{22} \\
 & \text{ }24 \\
 & \underline{\text{ }22} \\
 & \text{ 22} \\
 & \underline{\text{ 22}} \\
 & \text{ 0} \\
\end{align}}\right.}}\]
So, the quotient is 111……(ii)
(3) When divided by 37
The sum of the number is 2442. Dividing by 37, we have
\[37\overset{66}{\overline{\left){\begin{align}
  & 2442 \\
 & \underline{222} \\
 & \text{ }222 \\
 & \text{ }\underline{222} \\
 & \text{ 0} \\
\end{align}}\right.}}\]
So, the quotient is 66….(iii)
Therefore, from equations (i), (ii), and (iii), we have the quotients as
\[22,111,66\]
Hence, option (b) is the right answer.

Note:
The tricky part in this question was at the start of the question itself where we have to form two more numbers using 985 in cyclic order. The number 589 is not possible. As we need the order \[9\to 8\to 5\] and \[8\to 9\to 5,\] 9 goes to 8, \[\Rightarrow 8\to 9\] is wrong. Hence all other possibilities are ignored.