Smallest positive integer

We want to find the smallest positive $N$ such that $N = 2007a = 10000b + 2008$ for integers $a,b$ . Since $1 = 172 \times 10000 - 857 \times 2007$ , we need to solve $\begin{aligned} 2007a - 10000b & = \; 2008(172 \times 10000 - 857 \times 2007) \\ 2007(a + 2008 \times 857) & = \; 10000(b + 2008 \times 172) \end{aligned}$ and so (since $2007$ and $10000$ are coprime) we deduce that $a \; = \; 10000c - 857\times2008 \hspace{2cm} b \; = \; 2007c - 172\times2008 \hspace{3cm} c \in \mathbb{N}$ We need the smallest possible positive value of $a = 10000c - 1720856$ , and we do this by choosing $c = 173$ , obtaining $a = 9144$ , and hence $N = 2007a = \boxed{18352008}$ .

Let N be the desired integer. Since we need last four digits to be 2008, 2007 * N=#####4, since 4 * 8=42.
So 2007 * 4= 8028. To convert 2 in the tenth place to 0. we must add 80. (7 * 4=#8). This again comes from a 4. That is N=####44.
S0 2007 * 44= 88308. To convert 3 in the 10^2 th place to 0, we must add 700. (7 * 1=7). This again comes from a 1. That is N=###144
S0 2007 * 144= 289008. To convert 9 in the 10^3 th place to 2, we must add 3000. (7 * 9=#3). This again comes from a 9. That is N=###9144
S0 2007 * 9144= 18352008. We have done with smallest integer since we are approaching from digit with the smallest value.
The integer is $\Large \color{#D61F06}{18352008} .$

I set up the equation: 2007K=xx...x2008. I solved this problem by finding K and then multiplying by 2007. We know that the last digit of K has to be 4 because this is the only single digit number that when multiplied by 7 (the last digit of 2007) ends in 8 (the last digit of 2008). Then I multiplied 2000 and 4 (the already found last digit of K) to get 8000. 7K + 8000 must then have 2008 as its last four digits because the 2000 part of 2007 no longer contributes to the last four digits beyond its multiplication by the rightmost digit of K, 4. And so for in order for 2007K to end in 2008, 7K must end in 2008 - 8000 = 4008 (mod 10000). So, I first try to see if there exists an n such that 7K = 10000n + 4008, hopeful that 7K is a 5 digit number. If there does, then we are done (spoiler: there does). Since 10000/7 has a remainder of 4 and 4008/7 has a remainder of 4, then we must have 4n + 4 = 0 (mod 7). The smallest positive integer value of n for which this is achieved is n = 6. Therefore, 7K=64008 or K = 9144. Resulting in the final answer of 2017K = 18352008.