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 $\square$. (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 $\square$. (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": [ "Prime Factorization", "Least Common Multiple" ], "new_concepts": [ "Greatest Common Divisor from Prime Factorization" ], "current_concepts": [ "Prime Factorization", "Least Common Multiple", "Greatest Common Divisor from Prime Factorization" ] } ## Identify the prime factorizations Using the **Prime Factorization** knowledge point $$ \begin{aligned} r &= 5^3 \cdot 7^2 \cdot 11^1 \ s &= 5^4 \cdot 7^1 \cdot 11^2 \end{aligned} $$ ## Determine the minimum exponents for GCD To find the greatest common divisor (GCD), we take the minimum exponent for each common prime factor: $$ \begin{aligned} \min(3, 4) &= 3 \quad \text{for prime } 5 \ \min(2, 1) &= 1 \quad \text{for prime } 7 \ \min(1, 2) &= 1 \quad \text{for prime } 11 \end{aligned} $$ ## Calculate the final GCD value Using the **Least Common Multiple** knowledge point $$ \begin{aligned} \text{GCD} &= 5^3 \cdot 7^1 \cdot 11^1 \ &= 125 \cdot 7 \cdot 11 \ &= 875 \cdot 11 \ &= 9625 \end{aligned} $$ 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.) { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Number Theory", "Greatest Common Divisor" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the prime factorizations

Determine the minimum exponents for GCD

Calculate the final GCD value

Answer: