1. **Problem Statement:** Find the least common multiple (LCM) of the given numbers and algebraic expressions. 2. **Recall the formula and rules:** - The LCM of numbers is the smallest positive integer divisible by all the numbers. - For numbers, find prime factorizations and take the highest powers of all primes. - For algebraic terms, take the highest powers of each variable and coefficients' LCM. --- ### a) Find LCM of 32, 48, and 60 3. Prime factorize each: - $32 = 2^5$ - $48 = 2^4 \times 3$ - $60 = 2^2 \times 3 \times 5$ 4. Take highest powers of each prime: - For 2: highest power is $2^5$ - For 3: highest power is $3^1$ - For 5: highest power is $5^1$ 5. Multiply these: $$\text{LCM} = 2^5 \times 3 \times 5 = 32 \times 3 \times 5 = 480$$ --- ### b) Find LCM of $4x^2 y^5$ and $6x^3 y^2$ 6. Factor coefficients: - 4 = $2^2$ - 6 = $2 \times 3$ 7. Variables: - For $x$: highest power is $x^3$ - For $y$: highest power is $y^5$ 8. LCM of coefficients: - Prime factors combined: $2^2$ (from 4) and $3^1$ (from 6) - So LCM coefficient = $2^2 \times 3 = 4 \times 3 = 12$ 9. Combine coefficients and variables: $$\text{LCM} = 12 x^3 y^5$$ --- **Final answers:** - a) LCM = 480 - b) LCM = $12 x^3 y^5$