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Question
Given that the number \[\overline{{35\alpha64}}\] is divisible by 3, where α is a digit, what are the possible values of α?
Solution
\[\text{ It is given that } \overline{{35a64}} \text{ is a multiple of 3 }. \]
\[ \therefore (3 + 5 + a + 6 + 4) \text{ is a multiple of 3 }. \]
\[ \therefore (a + 18)\text{ is a multiple of 3} . \]
\[ \therefore (a + 18) = 0, 3, 6, 9, 12, 15, 18, 21 . . . \]
\[\text{ But a is a digit of number }\overline{{35a64}} \text{. So, a can take value }0, 1, 2, 3, 4 . . . 9 . \]
\[a + 18 = 18 \Rightarrow a = 0\]
\[a + 18 = 21 \Rightarrow a = 3\]
\[a + 18 = 24 \Rightarrow a = 6\]
\[a + 18 = 27 \Rightarrow a = 9\]
\[ \therefore a = 0, 3, 6, 9\]
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