The number of 3-digit odd numbers, whose sum of digits is a multiple of 7, is _____________.

For odd number unit place shall be $1,3,5,7$ or 9 .<br/><br/> $\therefore$ x y 1, x y 3, x y 5, x y 7, x y 9 are the type of numbers. numbers.<br/><br/> If $x \,y\, 1$ then $x+y=6,13,20$... Cases are required<br/><br/> i.e., $6+6+0+\ldots=12$ ways<br/><br/> If $x \, y \,3$ then<br/><br/> $x+y=4,11,18, \ldots$ Cases are required<br/><br/> i.e., $4+8+1+0 \ldots=13$ ways<br/><br/> Similarly for $x \,y \,5$, we have $x+y=2,9,16, \ldots$<br/><br/> i.e., $2+9+3=14$ ways<br/><br/> for $x \,y$ we have<br/><br/> $x+y=0,7,14, \ldots$<br/><br/> i.e., $0+7+5=12$ ways<br/><br/> And for $x$ y we have<br/><br/> $x+y=5,12,19 \ldots$<br/><br/> i.e., $5+7+0 \ldots=12$ ways<br/><br/> $\therefore$ Total 63 ways