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Math 55, Euclidean Algorithm Worksheet Feb 12, 2013 For each pair of integers (a, b), use the Euclidean algorithm to find their CD. Then reverse the steps of the algorithm to find integers s and t.
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Integers FAQ
When considering the positive integers 3054 and 12378, for instance, we found that gcd(3054, 12378)=6; whence, lcm(3054,12378)= 3054·12378 /6 =6300402.
Example: Express GCD(662, 414) = 2 as a linear combination of 662 and 414. Example: Express GCD(662, 414) = 2 as a linear combination of 662 and 414. a | bc, then a | c. The generalized Lemma can be used in the proof of uniqueness of prime factorizations.
Yes, therefore we are done – we have found the greatest common divisor and it is 6, hence, gcd(12, 18) = 6. The first step is to break each number into its prime factorization, then discern all the factors the two numbers have in common. Multiply these together. The result is the greatest common divisor.
The Euclidean Algorithm for finding GCD(A,B) is as follows: If A = 0 then GCD(A,B)=B, since the GCD(0,B)=B, and we can stop. If B = 0 then GCD(A,B)=A, since the GCD(A,0)=A, and we can stop. Write A in quotient remainder form (A = B⋅Q + R) Find GCD(B,R) using the Euclidean Algorithm since GCD(A,B) = GCD(B,R)
The Euclidean Algorithm for finding GCD(A,B) is as follows: If A = 0 then GCD(A,B)=B, since the GCD(0,B)=B, and we can stop. If B = 0 then GCD(A,B)=A, since the GCD(A,0)=A, and we can stop. Write A in quotient remainder form (A = B⋅Q + R) Find GCD(B,R) using the Euclidean Algorithm since GCD(A,B) = GCD(B,R)
When 54321 is divided by 12345, the quotient is 4 and the remainder is 4941: 54321=(4×12345)+4941. Therefore (as Euclid taught us), gcd(12345,54321)=gcd(12345,4941).
The Euclidean algorithm is a way to find the greatest common divisor of two positive integers, a and b. First let me show the computations for a=210 and b=45. Divide 210 by 45, and get the result 4 with remainder 30, so 210=4·45+30. Divide 45 by 30, and get the result 1 with remainder 15, so 45=1·30+15.
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