A list of 2018 positive integers has a unique mode, which occurs exactly 10 times. What is the least number of distinct values that can occur in the list?

234
224
223
202
225

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The mode appears 10 times, so there are $2018-10=2008$ numbers left to distribute among other values.

To get the smallest number of distinct values, we need to have each appear as many times as possible, and that is 9 times at most.

$\frac{2008}9=223+\frac19$

This can be covered by $223+1=224$ values.

Adding the mode makes a total of $\boxed{225}$ values.