We want just the last directory in the tree from this list:

# tail -3 list
/mnt/fs92/vol3/users/520/520/1680520 -- 8631
/mnt/fs92/vol3/users/568/8568/1578568 -- 2
/mnt/fs92/vol3/users/429/7429/1757429 -- 2

You can use both cut and awk, or awk twice…
weak:

# tail -3 list | cut -d / -f 8 | awk '{print $1}'
1680520
1578568
1757429

Or you can just tell awk to use multiple field separators. They are bound by square brackets, and in this case we can use both / and a space to be the input field separators, making the 8th column the one we are interested in.

strong:

# tail -3 list | awk -F "[/ ]" '{print $8}'
1680520
1578568
1757429