Math Problem Statement
Find the greatest common divisor of 2947 and 3997. Indicate the property, definition, or theorem
Solution
To find the Greatest Common Divisor (GCD) of 2947 and 3997, we can use the Euclidean algorithm. This algorithm repeatedly applies the division algorithm to find the GCD. The key property used here is:
Theorem (Euclidean Algorithm):
For two integers and where , the greatest common divisor of and is the same as the greatest common divisor of and . That is:
We repeat this process until one of the numbers becomes 0, at which point the GCD is the other number.
Now let's apply the Euclidean algorithm to 2947 and 3997.
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gives quotient 1, remainder .
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gives quotient 2, remainder .
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gives quotient 1, remainder .
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gives quotient 4, remainder .
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gives quotient 5, remainder .
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gives quotient 1, remainder .
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gives quotient 4, remainder .
Thus, the GCD of 2947 and 3997 is 7.
Would you like to see any additional steps or explanations?
Here are some related questions:
- Can the Euclidean algorithm be used for more than two numbers?
- How do you find the least common multiple (LCM) using the GCD?
- What is the relationship between the GCD and prime factorization?
- How does the extended Euclidean algorithm work?
- What is the GCD of two prime numbers?
Tip: The Euclidean algorithm is a fast and efficient method for computing the GCD, even for large numbers.
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Math Problem Analysis
Mathematical Concepts
Number Theory
Greatest Common Divisor (GCD)
Formulas
Euclidean Algorithm: gcd(a, b) = gcd(b, a mod b)
Theorems
Euclidean Algorithm
Suitable Grade Level
Grades 9-12
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