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 $a$ ?

Accepted Answer

E. The 4-digit number $N = 1000a + 111b$. For $N$ to be divisible by $a$, $1000a + 111b \equiv 0 \pmod a \iff 111b \equiv 0 \pmod a$. Since $111 = 3 	imes 37$, $111b = 37 	imes 3b$. Since $a$ is a single digit ($1 \le a \le 9$), $a$ is coprime to 37. Therefore, $37 	imes 3b$ is divisible by $a$ if and only if $3b$ is divisible by $a$.

Divisibility Condition for abbb

Explanation

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