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  • The number of 3-digit odd numbers, whose sum of digits is a multiple of 7, is__________.Option: 1 63Opt

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

Option: 1

63


Option: 2

-


Option: 3

-


Option: 4

-


Answers (1)

best_answer

Largest digit in number \mathrm{= 9}
Sum of three digit maximum can be \mathrm{= 27}
Hence sum of digit can be \mathrm{=7,14,21}

\mathrm{\; \; a_{1}+a_{2}+a_{3}=7,14,21}

\mathrm{case \,\,1: a_{1}+a_{2}+a_{3}=7 \quad a_{3}= odd }
\mathrm{a_{3}=1 \quad\left(a_{1}, a_{2}\right) \equiv(1,5),(2,4),(3,3),(4,2),(5,1),(6,0): Total\; number = 6}
\mathrm{a_{3}=3 \quad\left(a_{1}, a_{2}\right) \equiv(1,3),(2,2),(3,1),(4,0): Total \; number =4}
\mathrm{a_{3}=5\,\,\,\left(a_{1}, a_{2}\right) \equiv(1,1), (2,0): Total\: number =2}

\mathrm{case -2 \quad a_{1}+a_{2}+a_{3}=14}
\mathrm{a_{3}=1 \quad\left(a_{1}, a_{2}\right) \equiv(4,9),(5,8), \ldots .(9,4) :Total \: number =6}
\mathrm{a_{3}=3 \quad\left(a_{1}, a_{2}\right) \equiv(2,9),(3,8) \cdots(9,2) :Total \, number =8}
\mathrm{a_{3}=5 \quad\left(a_{1}, a_{2}\right) \equiv(1,8) \cdots \cdots(8,1) ,(9,0):Total \, number =9}
\mathrm{a_{3}=7 \quad\left(a_{1}, a_{2}\right) \equiv(1,6) \ldots(6,1),(7,0) :Total \, number =7}
\mathrm{a_{3}=9 \quad\left(a_{1}, a_{2}\right) \equiv(1,4) \ldots(4,1) ,(5,0):Total \, number =5}

\mathrm{case3\Rightarrow \quad a_{1}+a_{2}+a_{3}=21}
\mathrm{a_{3}=3 \quad\left(a_{1}, a_{2}\right) \equiv(9,9): Total \, number =1}
\mathrm{a_{3}=5\quad \left(a_{1}, a_{2}\right) \equiv(7,9),(8,8),(9,7): Total \, number =3}
\mathrm{a_{3}=7 \quad\left(a_{1}, a_{2}\right) \equiv(5,9) \ldots .(9,5): Total \, number =5}
\mathrm{a_{3}=9 \quad\left(a_{1}, a_{2}\right) \equiv(3,9) \cdots(9,3) :Total\, number =7}

\mathrm{Total =63}

Posted by

sudhir.kumar

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