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: Express the number algebraically. The number abbb can be represented as: N = 1000a + 100b + 10b + b = 1000a + 111b.

Step 2: Determine divisibility by a. For N to be divisible by a, the remainder must be 0:

N = 1000a + 111b. Since 1000a is divisible by a, 111b must also be divisible by a.

Step 3: Simplify the divisibility requirement. The condition 111b being divisible by a implies that b must be divisible by a.

Answer: Option 2.