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?

abbb can be written as (1000*a+100*b+10*b+1*b) = 1000a+111b

Now, we need to check to the necessary and sufficient condition for which (1000a+111b) is divisible by a.

We know that 1000a is always divisible by 'a', hence, we need to check for which condition, 111b is always divisible by a.

111b can be written as (3*37*b) => (3b*37) must be divisible by a.

a can't be a factor of 37, which implies 'a' is a factor of 3b => 3b is divisible by 'a'

The correct option is E