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Question
Given that the number \[\overline{{67y19}}\] is divisible by 9, where y is a digit, what are the possible values of y?
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Solution
\[\text{ It is given that }\overline{{67y19}}\text{ is a multiple of 9 }. \]
\[ \therefore (6 + 7 + y + 1 + 9)\text{ is a multiple of 9 .} \]
\[ \therefore (23 + y)\text{ is a multiple of 9 . }\]
\[23 + y = 0, 9, 18, 27, 36 . . . \]
\[\text{ But }x\text{ is a digit . So, }x\text{ can take values }0, 1, 2, 3, 4 . . . 9 . \]
\[23 + y = 27\]
\[ \Rightarrow y = 4\]
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