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- https://math.libretexts.org/Bookshelves/PreAlgebra/Prealgebra_(Arnold)/04%3A_Fractions/4.02%3A_Equivalent_FractionsIn this section we deal with fractions, numbers or expressions of the form a/b.
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Book%3A_Mathematical_Reasoning__Writing_and_Proof_(Sundstrom)/08%3A_Topics_in_Number_Theory/8.01%3A_The_Greatest_Common_DivisorOne of the most important concepts in elementary number theory is that of the greatest common divisor of two integers. Let a and b be integers, not both 0. A common divisor of a and b is any n...One of the most important concepts in elementary number theory is that of the greatest common divisor of two integers. Let a and b be integers, not both 0. A common divisor of a and b is any nonzero integer that divides both a and b . The largest natural number that divides both a and b is called the greatest common divisor of a and b .
- https://math.libretexts.org/Courses/Mount_Royal_University/Higher_Arithmetic/4%3A_Greatest_Common_Divisor_least_common_multiple_and_Euclidean_Algorithm/4.1%3A_Greatest_Common_DivisorThe greatest common divisor of two integers, also known as GCD, is the greatest positive integer that divides the two integers.
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Gentle_Introduction_to_the_Art_of_Mathematics_(Fields)/03%3A_Proof_Techniques_I/3.06%3A_Proofs_and_Disproofs_of_Existential_StatementsFrom a certain point of view, there is no need for the current section. If we are proving an existential statement we are disproving some universal statement. (Which has already been discussed.) Simil...From a certain point of view, there is no need for the current section. If we are proving an existential statement we are disproving some universal statement. (Which has already been discussed.) Similarly, if we are trying to disprove an existential statement, then we are actually proving a related universal statement. Nevertheless, sometimes the way a theorem is stated emphasizes the existence question over the corresponding universal.
- https://math.libretexts.org/Courses/Monroe_Community_College/MTH_220_Discrete_Math/3%3A_Proof_Techniques/3.5%3A_The_Euclidean_AlgorithmOne of the most important concepts in elementary number theory is that of the greatest common divisor of two integers. Let a and b be integers, not both 0. A common divisor of a and b is any n...One of the most important concepts in elementary number theory is that of the greatest common divisor of two integers. Let a and b be integers, not both 0. A common divisor of a and b is any nonzero integer that divides both a and b . The largest natural number that divides both a and b is called the greatest common divisor of a and b .
- https://math.libretexts.org/Courses/Western_Technical_College/PrePALS_PreAlgebra/02%3A_Fractions/2.01%3A_Properties_of_Fractions_and_Reducing_to_Lowest_TermsIn this section we deal with fractions, numbers or expressions of the form a/b.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Yet_Another_Introductory_Number_Theory_Textbook_-_Cryptology_Emphasis_(Poritz)/01%3A_Well-Ordering_and_Division/1.05%3A_The_Greatest_Common_DivisorIn this section we define the greatest common divisor (gcd) of two integers and discuss its properties. We also prove that the greatest common divisor of two integers is a linear combination of these ...In this section we define the greatest common divisor (gcd) of two integers and discuss its properties. We also prove that the greatest common divisor of two integers is a linear combination of these integers.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/A_Spiral_Workbook_for_Discrete_Mathematics_(Kwong)/05%3A_Basic_Number_Theory/5.04%3A_Greatest_Common_DivisorsIf we denote \(b=r_0\) and \(a=r_1\), then \[\begin{array}{rcl@{\qquad\qquad}l} r_0 &=& r_1 q_1 + r_2, & 0\leq r_2 < r_1, \\ r_1 &=& r_2 q_2 + r_3, & 0\leq r_3 < r_2, \\ r_2 &=& r_3 q_3 + r_4, & 0\leq...If we denote \(b=r_0\) and \(a=r_1\), then \[\begin{array}{rcl@{\qquad\qquad}l} r_0 &=& r_1 q_1 + r_2, & 0\leq r_2 < r_1, \\ r_1 &=& r_2 q_2 + r_3, & 0\leq r_3 < r_2, \\ r_2 &=& r_3 q_3 + r_4, & 0\leq r_4 < r_3, \\ \vdots & & \vdots \\ r_{k-1} &=& r_k q_k + r_{k+1}, & 0\leq r_{k+1} < r_k, \\ \vdots & & \vdots \\ r_{n-3} &=& r_{n-2} q_{n-2} + r_{n-1}, & 0\leq r_{n-1} < r_{n-2}, \\ r_{n-2} &=& r_{n-1} q_{n-1} + r_n, & r_n=0. \end{array} \nonumber\] It follows that \[\gcd(b,a) = \gcd(r_0,r_1) = \g…
- https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/Discrete_Structures/08%3A_Topics_in_Number_Theory/8.01%3A_The_Greatest_Common_DivisorOne of the most important concepts in elementary number theory is that of the greatest common divisor of two integers. Let a and b be integers, not both 0. A common divisor of a and b is any n...One of the most important concepts in elementary number theory is that of the greatest common divisor of two integers. Let a and b be integers, not both 0. A common divisor of a and b is any nonzero integer that divides both a and b . The largest natural number that divides both a and b is called the greatest common divisor of a and b .
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Number_Theory_(Raji)/01%3A_Introduction/1.05%3A_The_Greatest_Common_DivisorIn this section we define the greatest common divisor (gcd) of two integers and discuss its properties. We also prove that the greatest common divisor of two integers is a linear combination of these ...In this section we define the greatest common divisor (gcd) of two integers and discuss its properties. We also prove that the greatest common divisor of two integers is a linear combination of these integers.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Number_Theory_(Barrus_and_Clark)/01%3A_Chapters/1.07%3A_Greatest_Common_Divisor_and_Least_Common_MultipleIn the last few chapters we have discussed divisibility and the Division Algorithm when a single number is divided by another. In this chapter we begin to look at divisors and multiples that two numbe...In the last few chapters we have discussed divisibility and the Division Algorithm when a single number is divided by another. In this chapter we begin to look at divisors and multiples that two numbers have in common.
