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- 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/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/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/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=r0 and a=r1, 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=r0 and a=r1, then r0=r1q1+r2,0≤r2<r1,r1=r2q2+r3,0≤r3<r2,r2=r3q3+r4,0≤r4<r3,⋮⋮rk−1=rkqk+rk+1,0≤rk+1<rk,⋮⋮rn−3=rn−2qn−2+rn−1,0≤rn−1<rn−2,rn−2=rn−1qn−1+rn,rn=0. 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 .
