Get index of data.table column that matches a value

DT = data.table(
      id = 1:5,
      a  = c(0,1,0,2,5),
      b  = c(1,0,2,4,4),
      c  = c(1,2,0,0,5))

#     id a b c
# 1:  1  0 1 1
# 2:  2  1 0 2
# 3:  3  0 2 0
# 4:  4  2 4 0
# 5:  5  5 4 5

I want to identify the first column from the left which has 0, and put the column index in idx.

#     id a b c idx
# 1:  1  0 1 1 2 
# 2:  2  1 0 2 3
# 3:  3  0 2 0 2
# 4:  4  2 4 0 4
# 5:  5  5 4 5 NA

(Non-data.table solutions, e.g., with dplyr are also welcome)