#Pyth, 8 bytes
Pyth, 8 bytes
fq`;{`*Q #How?
How?
fq`;{`*Q - Full program. f - First input where the condition is truthy over [1, ∞) *Q - The product of the input and the current number ` - Converted to String. { - Deduplicated. q - Is equal to? `; - The string representation of the innermost lambda variable, T, which is "10". - Output implicitly. This pretty much relies on the fact that numbers do not have trailing zeros, and 1 always occurs before 0 in a positive integer only consisting of 0s and 1s.