JEE MAIN - Mathematics (2022 - 25th June Morning Shift - No. 16)
The number of 3-digit odd numbers, whose sum of digits is a multiple of 7, is _____________.
Answer
63
Explanation
For odd number unit place shall be $1,3,5,7$ or 9 .
$\therefore$ x y 1, x y 3, x y 5, x y 7, x y 9 are the type of numbers. numbers.
If $x \,y\, 1$ then $x+y=6,13,20$... Cases are required
i.e., $6+6+0+\ldots=12$ ways
If $x \, y \,3$ then
$x+y=4,11,18, \ldots$ Cases are required
i.e., $4+8+1+0 \ldots=13$ ways
Similarly for $x \,y \,5$, we have $x+y=2,9,16, \ldots$
i.e., $2+9+3=14$ ways
for $x \,y$ we have
$x+y=0,7,14, \ldots$
i.e., $0+7+5=12$ ways
And for $x$ y we have
$$ x+y=5,12,19 \ldots $$
i.e., $5+7+0 \ldots=12$ ways
$\therefore$ Total 63 ways
$\therefore$ x y 1, x y 3, x y 5, x y 7, x y 9 are the type of numbers. numbers.
If $x \,y\, 1$ then $x+y=6,13,20$... Cases are required
i.e., $6+6+0+\ldots=12$ ways
If $x \, y \,3$ then
$x+y=4,11,18, \ldots$ Cases are required
i.e., $4+8+1+0 \ldots=13$ ways
Similarly for $x \,y \,5$, we have $x+y=2,9,16, \ldots$
i.e., $2+9+3=14$ ways
for $x \,y$ we have
$x+y=0,7,14, \ldots$
i.e., $0+7+5=12$ ways
And for $x$ y we have
$$ x+y=5,12,19 \ldots $$
i.e., $5+7+0 \ldots=12$ ways
$\therefore$ Total 63 ways
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