Question

S is a set of five-digit numbers formed using the natural numbers between 2 and 8 exactly once. How many numbers of set S are divisible by 11?

• A. 9
• B. 10
• C. 11
• D. 12
• E. 13

Answer 1

The Correct Option is D

Let \(‘abcde’\) be a five-digit number.
The numbers between \(2\) and \(8\) are \(3, 4, 5, 6\), and \(7\).
For the number to be divisible by \(11, |(a + c + e) – (b + d)|\) = Multiple of \(11\)
There are three possible cases.

Case 1: \((a + c + e) – (b + d) = 0\)
or, \((a + c + e) + (b + d) = 2(b + d)\)
or, \(25 = 2(b + d)\)
This is not possible as \(b + d \) must be an integer.

Case 2: \((a + c + e) – (b + d) = 11\)
\((a + c + e) – (b + d) = 11\)
or, \((a + c + e) + (b + d) = 2 \times (b + d) + 11\)
or, \(7 = (b + d)\)
There are two possibilities, \((3, 4)\) and \((4, 3)\).
The remaining three digits can be arranged in \(3! = 6\) ways.
Thus, \(2\times6 = 12\) ways

Case 3: \((a + c + e) – (b + d) = –11\)
\((a + c + e) – (b + d) = –11\)
or, \((a + c + e) + (b + d) = 2 x (b + d) – 11\)
or, \(18 = (b + d)\)
This is not possible, as the maximum possible sum of the two digits is \(13\).
Therefore, the total numbers in Set S that are divisible by \(11\) is \(12\).

Hence, option D is the correct answer.