1. **Problem:** Determine if the ISBN number 978-1-4116-8691-5 is valid. 2. **Formula and rules:** ISBN-13 validation uses the formula: $$\text{Sum} = \sum_{i=1}^{13} d_i \times w_i$$ where $d_i$ are the digits and weights $w_i$ alternate between 1 and 3 starting with 1 for the first digit. The ISBN is valid if $\text{Sum} \equiv 0 \pmod{10}$. 3. **Digits and weights:** Digits: 9,7,8,1,4,1,1,6,8,6,9,1,5 Weights: 1,3,1,3,1,3,1,3,1,3,1,3,1 4. **Calculate weighted sum:** $$9\times1=9$$ $$7\times3=21$$ $$8\times1=8$$ $$1\times3=3$$ $$4\times1=4$$ $$1\times3=3$$ $$1\times1=1$$ $$6\times3=18$$ $$8\times1=8$$ $$6\times3=18$$ $$9\times1=9$$ $$1\times3=3$$ $$5\times1=5$$ Sum all: $$9+21+8+3+4+3+1+18+8+18+9+3+5=110$$ 5. **Check validity:** $$110 \mod 10 = 0$$ Since the remainder is 0, the ISBN is valid. --- **Final answer:** The ISBN 978-1-4116-8691-5 is valid.