Let the three-digit odd number be xyz
Now z=1/3/5/7/9
x+y+z=7/14/21 [for sum of digit to be multiple of 7]
x+y=6/4/2/13/11/9/7/5/20/18/16/14/12
When x+y=6⇒(1,5),(2,4),(3,3),(4,2),(5,1),(6,0)
Total possibilities=6
When x+y=4⇒(1,3),(2,2),(3,1),(4,0)
Total possibilities=4
When x+y=2⇒(1,1),(2,0)
Total possibilities=2
When x+y=13⇒(4,9),(5,8),(6,7),(7,6),(8,5),(9,4)
Total possibilities=6
When x+y=11⇒(2,9),(3,8),(4,7),(5,6),(6,5),(6,5),(7,4),(8,3),(9,2)
Total possibilities=8
When x+y=9⇒(1,8),(2,7),(3,8),(4,5),(5,4),….(8,1),(9,0)
Total possibilities=9
When x+y=7⇒(1,8),(2,5),(3,4),….(8,1),(7,0)
Total possibilities=7
When x+y=5⇒(1,4),(2,3),(3,2),(4,1),(5,0)
Total possibilities=5
When x+y=20 No possibilities
When x+y=18⇒(9,9)
Total possibilities=1
When x+y=16⇒(7,9),(8,8),(9,7)
Total possibilities=3
When x+y=14⇒(5,9),(6,8),(7,7),(8,6),(9,5)
Total possibilities=5
When x+y=12⇒(3,9),(4,8),(5,7),(6,6)….(9,3)
Total possibilities=7
Hence, the total number of three digit odd numbers whose sum of digit is divisible by 7 is 63
