Onnnnnn.....nnnnes

Relevant wiki: Application of Divisibility Rules

Due to the given condition, $R.H.S$ should be divisible by $99$ . $\begin{aligned} \text{Hence, 9 | 111} \ldots \text{1} \\ \text{ Also, 11 | 111}\ldots \text{1} \end{aligned}$ Therefore, the number of 1's must be divisible by 9. Such smallest possible value is $\underbrace{111 \ldots1}_{\text{ Nine 1's }}$ , but we note that this number is not divisible by 11 , because the number of digits are odd , for it to be perfectly divisible by 11 , the number of digits should by even.

Hence, the number should have 18 1's and hence $Y=\boxed{1122334455667789}$ .

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Note that the reasoning for divisibility of 11 holds here for this special case only.

99x112233445566778899=11111111111111111111