1. Problem: Find the Least Common Multiple (LCM) of 105 and 637 using their prime factor trees. 2. Prime factorization: - 105 = 3 \times 5 \times 7 - 637 = 7 \times 7 \times 13 = 7^{2} \times 13 3. To find the LCM, take the highest powers of all prime factors present: - For 3: $3^{1}$ (only in 105) - For 5: $5^{1}$ (only in 105) - For 7: $7^{2}$ (since 7 appears squared in 637) - For 13: $13^{1}$ (only in 637) 4. Multiply these together: $$ \text{LCM} = 3 \times 5 \times 7^{2} \times 13 $$ 5. Calculate the numerical value: - $7^{2} = 49$ - $3 \times 5 = 15$ - $15 \times 49 = 735$ - $735 \times 13 = 9555$ 6. Final answer: The LCM of 105 and 637 is $9555$.