What is the smallest eight-digit positive integer that has exactly four digits which are 4 ?
The smallest eight-digit number is 10000000 .

The smallest 10000 eight-digit numbers are those of the form $1000 a b c d$ where $a, b, c$, and $d$ are digits. Among the 10000 smallest eight-digit numbers, 10004444 is the only one that has four digits that are equal to 4.

This means the smallest eight-digit positive integer that has exactly four digits which are 4 is 10004444 .

ANSWER: 10004444