IMO: 63492
I dont know the shorter method: but I took less than a min.
wrote down table of
7, 14, 21, 28, 35, 42, 49, 56, 63, 70
need all fours
for units to be 4, the number x unit didigit must be 2 (as 2 x 7 = 14)
we carry forward 1, now the tens digit when multiplied by 7 should give 3, that is done by 9 so we have 63+1=64
carry forward 6
we require 8 in product as 8+6=14, so we take 4
4x7=28, 28+6=34
carry forward 3
we require 1. 3X7=21
21+3=24
carry forward 2. we require 2
6x7=42, 42+2=44
both 4
we construct the number
2 + 10*9 + 100*4 + 1000*3 +10000*6 = 63492