Question

find the least common multiple (lcm) of the following: 15. 45, 75, and 125 16. $5x^{5}y^{3}z$ and $6x^{3}y^{4}z^{2}$

Explanation:

Step1: Prime factorize each number

$45 = 3^2 \times 5^1$
$75 = 3^1 \times 5^2$
$125 = 5^3$

Step2: Take max exponents of primes

For prime 3: $\max(2,1,0) = 2$
For prime 5: $\max(1,2,3) = 3$

Step3: Calculate LCM of integers

$\text{LCM}(45,75,125) = 3^2 \times 5^3 = 9 \times 125 = 1125$

Step4: Identify terms for algebraic LCM

For $5x^5y^3z$ and $6x^3y^4z^2$:

• Coefficients: 5, 6
• $x$ exponents: 5, 3
• $y$ exponents: 3, 4
• $z$ exponents: 1, 2

Step5: Compute LCM of coefficients

$\text{LCM}(5,6) = 30$

Step6: Take max exponents of variables

$x^{\max(5,3)} = x^5$, $y^{\max(3,4)} = y^4$, $z^{\max(1,2)} = z^2$

Step7: Combine for algebraic LCM

$\text{LCM}(5x^5y^3z, 6x^3y^4z^2) = 30x^5y^4z^2$

Answer:

1. $1125$
2. $30x^5y^4z^2$