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

Alternate approach

increase 4s and check if the number is dvisible by 7

44 = no
444 = no
4444 = no
44444 = no
444444 = yes
quotient = 63492.

this method took more time than the previous one.

mental,
i followed the same approach as your second one. Dont know where i goofed up.......it should be correct ans.but i dont know the OA.
So, if anybody can come up sol lesser than that...we hv to c.

my first method took me less 60 sec.

though typing it was long.

The second method took more time and also there are more chances of error!!