Question

How many 4-digit numbers of the form AB61 are there that are divisible by 11 (where A and B are distinct digits)?

• A. 3
• B. 4
• C. 7
• D. 13
• E. 14

Answer 1

The Correct Option is B

To find how many 4-digit numbers of the form AB61 are divisible by 11, we start by applying the divisibility rule for 11. A number is divisible by 11 if the difference between the sum of its digits in odd positions and the sum of its digits in even positions is a multiple of 11.

The number is of the form AB61, where A and B are distinct digits. Therefore, the positions and their digits are:

• 1st position: A
• 2nd position: B 
• 3rd position: 6
• 4th position: 1

According to the divisibility rule of 11:

\(A + 6 - (B + 1)\) should be a multiple of 11.

 

Simplifying the condition:

\(A - B + 5 \equiv 0 \, (\text{mod} \, 11)\).

This simplifies to:

\(A - B \equiv -5 \, (\text{mod} \, 11)\) 
\(A - B \equiv 6 \, (\text{mod} \, 11)\)

 

Since A and B are digits (0-9), we solve:

• \(A - B = 6\)

We can find valid pairs (A, B):

• If A = 6, then B = 0 (6-0=6) ✔️
• If A = 7, then B = 1 (7-1=6) ✔️
• If A = 8, then B = 2 (8-2=6) ✔️
• If A = 9, then B = 3 (9-3=6) ✔️

Thus, the four valid numbers that can be formed are: 6061, 7161, 8261, and 9361.

Hence, there are 4 four-digit numbers of the form AB61 that are divisible by 11 where A and B are distinct digits.