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

Answer: (D)

12

Answer 2

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.