To multiply \( 2 \) by \( 1 \frac{2}{3} \), we first convert the mixed number \( 1 \frac{2}{3} \) into an improper fraction.
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Convert \( 1 \frac{2}{3} \) to an improper fraction: \[ 1 \frac{2}{3} = \frac{3 \times 1 + 2}{3} = \frac{3 + 2}{3} = \frac{5}{3} \]
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Now multiply \( 2 \) by \( \frac{5}{3} \): \[ 2 \times \frac{5}{3} = \frac{2 \times 5}{1 \times 3} = \frac{10}{3} \]
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Now convert \( \frac{10}{3} \) into a mixed number: \[ 10 \div 3 = 3 \quad \text{(integer part)} \] \[ 10 - (3 \times 3) = 10 - 9 = 1 \quad \text{(remainder)} \]
So, \( \frac{10}{3} = 3 \frac{1}{3} \).
Therefore, the answer is: \[ \boxed{3 \frac{1}{3}} \]
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