Question

Let $n$ denote the smallest positive integer that is divisible by both $4$ and $9,$ and whose base- $10$ representation consists of only $4$ 's and $9$ 's, with at least one of each. What are the last four digits of $n?$

$4444$

$4494$

$4944$

$9444$

$9944$

Answer: C

Concepts:divisibility digits optimization

Difficulty rating: 1980

Solution:

Since $n$ is divisible by $9,$ its digit sum is a multiple of $9.$ With $k$ fours and $m$ nines, the digit sum is $4k+9m,$ so $9\mid 4k,$ forcing $9\mid k.$ Thus $k\ge 9,$ and with at least one $9,$ the number has at least ten digits.

For divisibility by $4,$ the last two digits must form a multiple of $4,$ and among $44,49,94, 99$ only $44$ works, so $n$ ends in $44.$

The smallest such ten-digit number places the single $9$ in the lowest available position, giving $4{,}444{,}444{,}944.$ Its last four digits are $4944.$

Thus, the correct answer is C.

2007 AMC 10B Problem 24

Answer: C

Difficulty rating: 1980

Solution:

Problem 24 in Other Years