Question

If x is a digit of the number \[\overline {{66784x}}\] such that it is divisible by 9, find possible values of x.

Answer 1

\[\text{ It is given that }\bar{{66784x}}\text{ is a multiple of }9 . \]
\[\text{ Therefore, }(6 + 6 + 7 + 8 + 4 + x)\text{ is a multiple of }9 . \]
And,
\[(31 + x)\text{ is a multiple of 9 }. \]
\[\text{Possible values of }(31 + x) \text{ are }0, 9, 18, 27, 36, 45, . . . \]
\[\text{ But }x\text{ is a digit . So, }x \text{ can only take value }0, 1, 2, 3, 4, . . . 9 . \]
\[ \therefore 31 + x = 36 \]
\[ \Rightarrow x = 36 - 31\]
\[ \Rightarrow x = 5\]