Maybe I'm being too pedantic and missing the meaning in your post, yet to me, division is also asking, "What is the missing factor?" The following pedagogy may also work well for long division of arithmetic and algebraic expressions.
So the long division of 3680 divided by 46 is also asking us to find the missing product AB, that when multiplied by 46, equals 3680.
.......ThHTU solve for AB times......46 equals 3680
If we solve for A first...
Four doesn't go into 3, so how many times does 4 go into 36? (Just a conversation starter for children, before we include place value logic.)
NOTE: You can also ask, "What number A tens, times 4 tens will be as close as possible yet won't exceed 36 hundreds?" This gives 8 tens as if we say 9 tens, when we also cross multiply 9T x 6U we get more hundreds and we already have the 36 hundreds from our 9!
So 4 tens go into 360 tens safely 80 times with 40 tens as a Remainder (aka R)
So we get A = 8
Now solve for B units given that A tens = 80
Solve for .8B times.......46 equals..3680 less......3200 from 80 x 40 R 48T.....480
solve for B What must B units equal so that the cross products 8T x 6U plus 4T x BU = 48T (aka THE TOTAL REMAINDER)
As 8T x 6U = 48T already, B must equal zero units
AB = 80
====== Alternatively, using place value letters (Th, H, T, U) we can write it as...