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)

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

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

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

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

111b must be divisible by a.

This means b must be divisible by a.

Answer: Option 2.