39829

Nicholas’ 4 digit target solution:
My method:
First off I started with writing out all the numbers like this,
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
Next, I tried to think of the highest largest odd number possible and I thought of this 9867 and by doing this I realised I could do 9876 for the largest even number. With all of them gone I thought of multiples of three, and I saw 45 and 33 when I put them together I got 4533 but if I switch them round I get 5433 which is larger. The numbers I had left were, 5 0 0 1 1 2 2 4. Next I found out the smallest multiple of 5 which was, 1005 but I thought that that five could go somewhere else so came out with the second smallest number which was 1010. The numbers I had left were 2 2 4 5 and to get the closest to 5000 I got 5422. So in the end I got:
The largest even number: 9867
The largest odd number: 9876
Largest multiple of 3: 5433
Smallest multiple of 5: 1010
Closet to 5000: 4522