use the method based on prime-power representations to find the greatest common divisor and least common multiple of the given numbers.

$r = 5^3 \cdot 7^2 \cdot 11^1$ and $s = 5^4 \cdot 7^1 \cdot 11^2$

the gcd is 9625. (simplify your answer.)

the lcm is . (simplify your answer.)

Question

use the method based on prime-power representations to find the greatest common divisor and least common multiple of the given numbers.

$r = 5^3 \cdot 7^2 \cdot 11^1$ and $s = 5^4 \cdot 7^1 \cdot 11^2$

the gcd is 9625. (simplify your answer.)

the lcm is  . (simplify your answer.)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Greatest Common Divisor from Prime Factorization", "Least Common Multiple" ], "new_concepts": [], "current_concepts": [ "Greatest Common Divisor from Prime Factorization", "Least Common Multiple" ] } ## Identify prime-power representations Using the *Greatest Common Divisor from Prime Factorization* knowledge point $$ \begin{aligned} r &= 5^3 \cdot 7^2 \cdot 11^1 \ s &= 5^4 \cdot 7^1 \cdot 11^2 \end{aligned} $$ ## Calculate the Greatest Common Divisor (GCD) Using the *Greatest Common Divisor from Prime Factorization* knowledge point $$ \begin{aligned} \gcd(r, s) &= 5^{\min(3,4)} \cdot 7^{\min(2,1)} \cdot 11^{\min(1,2)} \ &= 5^3 \cdot 7^1 \cdot 11^1 \ &= 125 \cdot 7 \cdot 11 \ &= 9625 \end{aligned} $$ ## Calculate the Least Common Multiple (LCM) Using the *Least Common Multiple* knowledge point $$ \begin{aligned} \operatorname{lcm}(r, s) &= 5^{\max(3,4)} \cdot 7^{\max(2,1)} \cdot 11^{\max(1,2)} \ &= 5^4 \cdot 7^2 \cdot 11^2 \ &= 625 \cdot 49 \cdot 121 \ &= 30625 \cdot 121 \ &= 3705625 \end{aligned} $$ Use the method based on prime-power representations to find the greatest common divisor and least common multiple of $r$ and $s$. $r = 5^3 \cdot 7^2 \cdot 11^1$ and $s = 5^4 \cdot 7^1 \cdot 11^2$ The GCD is 9625. (Simplify your answer.) The LCM is 3705625. (Simplify your answer.) { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Number Theory", "Least Common Multiple" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Identify prime-power representations

Calculate the Greatest Common Divisor (GCD)

Calculate the Least Common Multiple (LCM)

Answer: