In the expression $10: 9: 8: 7: 6: 5: 4: 3: 2: 1$, parentheses were placed such that the value of the expression is an integer.

What is the smallest number that could have been obtained?

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In order for the value of the expression to be an integer, after placing the parentheses and writing the resulting expression as a common fraction, the number 7 must end up in the numerator. Therefore, the value of this expression is no less than 7. This can be achieved, for example, as follows:

$10: 9:(8: 7:(6:(5: 4:(3: 2: 1)))=(10 \cdot 7 \cdot 6 \cdot 4 \cdot 3):(9 \cdot 8 \cdot 5 \cdot 2 \cdot 1)=7$.

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