1. **State the problem:** Multiply 2 by the mixed number $2\frac{5}{8}$. 2. **Convert the mixed number to an improper fraction:** $$2\frac{5}{8} = \frac{2 \times 8 + 5}{8} = \frac{16 + 5}{8} = \frac{21}{8}$$ 3. **Write the multiplication:** $$2 \times \frac{21}{8}$$ 4. **Convert 2 to a fraction:** $$\frac{2}{1} \times \frac{21}{8}$$ 5. **Multiply the numerators and denominators:** $$\frac{2 \times 21}{1 \times 8} = \frac{42}{8}$$ 6. **Simplify the fraction by dividing numerator and denominator by their greatest common divisor (2):** $$\frac{\cancel{42}^{21}}{\cancel{8}^{4}} = \frac{21}{4}$$ 7. **Convert the improper fraction back to a mixed number:** $$\frac{21}{4} = 5 \frac{1}{4}$$ **Final answer:** $$5 \frac{1}{4}$$