Question

Consider a 4-digit number of the form abbb, i.e., the first digit is a (a > 0) and the last three digits are all b.
Which of the following conditions is both NECESSARY and SUFFICIENT to ensure that the 4- digit number is divisible by a?

• A. b is divisible by a
• B. b is equal to 0
• C. 21b is divisible by a
• D. 9b is divisible by a
• E. 3b is divisible by a

Answer 1

Answer: (E)

3b is divisible by a

Answer 2

Step 1: Write the number in terms of aand b. The number abbb can be expressed as:

N = 1000 a + 100 b + 10 b + b = 1000 a + 111 b.

Step 2: Condition for divisibility by a. For N to be divisible by a , the remainder when N is divided by a must be 0:

N = 1000 a + 111 b => 111 b must be divisible by a.

Step 3: Simplify the condition. Since 1000 a is always divisible by a , the divisibility condition reduces to:

111 b must be divisible by a.

This means b must be divisible by a.

Answer: Option 2.