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A mathematical method for finding the greatest common divisor (GCD) of two integers, based on the principles established by ancient Greek mathematician Euclid.
What is the main goal of the Euclidean Algorithm?
What is the common notation for denoting the GCD of two numbers using the Euclidean Algorithm?
=> write what the givens imply, create a general end goal statement (need to show statement)=> if you know that your ending statement needs to be in terms of a certain variable, use substition (example on page 5 proof of proposition 1.1)=> if we have two equations that equal 1, remember that multiplying them together will still result in 1 (homework problem like this, don't remember which)=> proof by cases (in chapter 1 the cases are mostly prime or composite since we are working with prime numbers/factorizations often, except that keeps popping up is in euclid's proof of infinite primes and ones based on his proof)=> euclidian's proof of infinite primes is important to understand since only his proof and dirichlet's theorem of infinite primes are the only ways to prove infinite primes=> when proving a number is composite, remember to factor and use the definition of composite (n,a,b ∈ *Z*, 1<a≤b<n for n=ab)(an example is 2²⁰ - 1 problem on the practice test) OR can factor using the bⁿ -1 theorem of the form (b - 1)(bⁿ-¹ + bⁿ-² ... b + 1) => WOP (set S to be a subset of the integers that you are exploring -with the OPPOSITE property of what you are trying to prove- that is nonempty and therefore has a least element, then prove by way of contradiction that... there's actually an element less than d that doesn't hold the property either =><=, or ... use an element that's out of S that's smaller than d to prove that d isn't actually in S (like in the stamp problem) =><=)=> induction (base case, induction step, induction hypothesis)=> when working with gcd, create an arbitrary variable that is the gcd to use throughout the proof (an example is the proof of proposition 1.10 on page 19)
let p ∈ *Z* with p>1. then p is said to be prime if the only positive divisors of p and 1 and p. if n ∈ *Z* with n>1 is not prime, then n is composite
let a,b ∈ *Z⁺*. the least common multiple of a and b [a,b] is the least positive integer m such that a|m and b|m**to find the lcm, find the prime factorization of both a and b, and find the maximum exponent on EACH prime from either a or bex. 8 has a pf of 2³, and 26 has a pf of 2*13. the lcm is 2³13 because the only primes are 2 and 13, and the maximum exponent of 2 is 3 in 8, and the maximum exponent of 13 is 1 in 26
Which ancient mathematician is believed to have used a precursor to the Euclidean Algorithm?
When using the Euclidean Algorithm, what happens when the remainder becomes zero?
What does the Euclidean Algorithm find for two identical numbers?
What is the GCD of 56 and 63 when using the Euclidean Algorithm?
What is the Euclidean Algorithm's space complexity?
What is the result of applying the Euclidean Algorithm to 48 and 18?
What is the Euclidean Algorithm's primary application in computer algorithms?
let x ∈ *R*. the greatest integer function of x ([x]) is the greatest integer less than or equal to x *proof of lemma 1.3 on page 6
What is the final step in each iteration of the Euclidean Algorithm?
any prime number expressible in the form 2²^ⁿ + 1 with n ∈ *Z* and n≥0 ex. 3 = 2²^⁰ +1, 5, 17, 257, 65537*there are said to be only 5 fermant primes
What is the Euclidean Algorithm's space complexity for finding the GCD of two numbers?
What does the Euclidean Algorithm find for coprime numbers?
there are infinitely many prime numbers*understanding this proof helps with other proofs
What is the GCD of 9 and 10 when using the Euclidean Algorithm?
What is the GCD of 63 and 35 when using the Euclidean Algorithm?
let x ∈ *R* with x > 0. then π(x) is the function defined |{p: p prime; 1<p≤x}| which can be estimated by he prime number theorem
What is the output of the Euclidean Algorithm for 12 and 7?
What is the primary application of the Euclidean Algorithm in cryptography?
every integer greater than 1 has a prime divisor
What is the next step after calculating the quotient in the Euclidean Algorithm?
How many dimensions does a surface have?
Who is the mathematician credited with the development of the Euclidean Algorithm?
let a, b ∈ *Z* with b > 0. then there exists unique q, r ∈ *Z* such that a = bq + r and 0≤r<b
What is the GCD of 42 and 56 when using the Euclidean Algorithm?
What does the Euclidean Algorithm find for two numbers with one of them being a decimal?
Which pair of numbers will result in a GCD of 1 when using the Euclidean Algorithm?
What is the worst-case time complexity of the Euclidean Algorithm?
any prime number expressible in the for 2ᵖ - 1 where p is prime is a mersenne primeex. 3 = 2²-1, 7, 31, 127, 8191
The three steps from solids to points are?
let a,b ∈ *Z⁺* and (a,b) =1. then a + b, a + 2b, a + 3b.... a + nb contains infinitely many primes**homework #3 problem like this, sample test problem
let a, b ∈ *Z*. if a|b and b|c, then a|c
Who is the founder of the Euclid geometry?
What is the GCD of 36 and 48 when using the Euclidean Algorithm?
What is the output of the Euclidean Algorithm for 17 and 11?
What property of numbers does the Euclidean Algorithm utilize to find the GCD?
let a,b ∈ *Z⁺*. the [a,b] = ab if and only if (a,b) = 1*this makes sense because since they're relatively prime, they have no factors in common. therefore, each multiple needs to have all the primes from each a and b, so [a,b] = ab
What is the output of the Euclidean Algorithm for 20 and 8?
Which operation is at the core of the Euclidean Algorithm?
let a, b, c, m, n ∈ *Z*. if c|a and c|b, then c|(ma + nb)*ma + nb is a integral linear combination of a and b
What does the Euclidean Algorithm find for two prime numbers?
lim π(x)lnx = 1x→∞ (x)
How many dimensions does a solid have?
What is the last step in the Euclidean Algorithm?
Which ancient civilization is closely associated with the early use of the Euclidean Algorithm?
What is the Euclidean Algorithm's main advantage in terms of computational efficiency?
Which step is repeated in the Euclidean Algorithm until a GCD is found?
when trying to find all prime numbers less than n, write all the numbers from 2-n (exclude 1, because 1 can't be prime) since every number less than n has a prime divisor less than √n, check for the prime numbers ≤√n and go through the list and cross out all multiples of that prime on the entire listthe numbers that aren't crossed out at the end are the prime numbersex. every number below or equal to 49 has a prime divisor p ≤√49 ≈ 7, so the prime numbers are 2,3,5, and 7. go through the list of 2-50 and cross out all numbers that are multiples of 2,3,5, and 7. the numbers that aren't crossed out are the prime numbers, which are 2,3,5,7,11,13,17,19,23,31,37,41,43 and 47.
Euclidean geometry is an axiomatic system, in which all theorems are derived from a small number of axioms.
What is the Euclidean Algorithm used for?
What is the next step after swapping the numbers in the Euclidean Algorithm?
The boundaries of solids are called?
for any positive integer n, there are at least n consecutive composite positive integers of the form (n+1)! + 2, (n+1)! + 3, ... (n+1)! + n + 1, or generally of the form (n+1)! + i where 2≤i≤n+1ex. the positive integer 4, there are at least 4 consecutive composite positive integers of the form (n+1)! + i where 2≤i≤5.(5)! + 2 = 122 (2*61)(5)! + 3 = 123 (3*41)(5)! + 4 = 124 (2*62)(5)! + 5 = 125 (5*25)
let a,b ∈ *Z* with a,b≠0. then (a,b) = min{ma + nb; n ∈ *Z*, ma+nb > 0}saying that the smallest positive value of the integral linear combination of a and b will be its gcd
n is a perfect square iff the exponents in the prime factors are all even
What is the primary purpose of the Euclidean Algorithm in computer science?
Which of the following is NOT a step in the Euclidean Algorithm?
let a,b,p ∈ *Z* with p being prime. if p|ab, then p|a or p|b
let a,b ∈ *Z* with a,b≠0. the greatest common divisor of a and b (a,b) is the greatest positive integer d such that d|a and d|b. if (a,b) = 1, then a and b are said to be relatively prime **to find the gcd, find the prime factorization of both a and b, and find the minimum exponent on BOTH primes from both a or bex. 8 has a pf of 2³, and 26 has a pf of 2*13. the gcd is 2 because the only prime they have in common is 2, and the minimum exponent of 2 is 1 in 26
What is the main purpose of the Euclidean Algorithm in number theory?
In which branch of mathematics is the Euclidean Algorithm extensively used?
What shape are side faces of a pyramid?
What is the alternative name for the Euclidean Algorithm?
What is the starting point of the Euclidean Algorithm?
there are infinitely many primes numbers p in which p+2 is also a prime number
What is the Euclidean Algorithm's worst-case time complexity when both numbers are equal?
What is the main advantage of the Euclidean Algorithm in terms of memory usage?
let a,b ∈ *Z* with (a,b) = d. then (a/d, b/d) = 1
see example in book
*summarizing because I'm not writing the definition as they have it down*if a, b, c..... n ∈ *Z* and are not all 0, and the gcd of (a, b, d... n) = 1, then a, b, c... n are relatively prime. if (a, b) = 1, (b, c) = 1... *all combinations of a, b, c... n*, then each pair is pairwise relatively prime as well
What is the final result of the Euclidean Algorithm for 14 and 9?
What is the result of using the Euclidean Algorithm for 56 and 48?
What is the GCD of 60 and 84 when using the Euclidean Algorithm?
keep in mind that the gcd of the two numbers you start with is the remainder that happens before the remainder hits 0
What is the fundamental principle behind the Euclidean Algorithm?
Which step follows calculating the quotient in the Euclidean Algorithm?
What is the Euclidean Algorithm's main limitation when dealing with floating-point numbers?
What is the result of applying the Euclidean Algorithm to 99 and 33?
every integer greater than 1 can be expressed in the form p sub 1≤i≤n with exponents a sub 1≤j≤n where the p's are distinct prime numbers and the a's are positive numbers. the prime factorization is unique except for the arrangement of the p's**when working with divisibility and exponents (a|b), remember that the exponents in the prime factorization of b are greater than a
What is the GCD of 33 and 14 when using the Euclidean Algorithm?
Hyperbolic geometry means the same thing as Euclidean geometry.
What is the first step in the Euclidean Algorithm?
What does the Euclidean Algorithm compute for two given numbers?
let a, b ∈ *Z*. a|b if there exists a c ∈*Z* such that b = ac
What is the Euclidean Algorithm's main advantage over other GCD calculation methods?
What is the GCD of 72 and 90 when using the Euclidean Algorithm?
What is the first step in each iteration of the Euclidean Algorithm?
What is the key idea in the Euclidean Algorithm for finding the GCD?
every even integer greater than 2 can be expressed as the sum of 2 (not necessarily distinct) primes
What does the Euclidean Algorithm find for two numbers with one of them being a fraction?
What is the output of the Euclidean Algorithm for 77 and 55?
How many dimensions does a point have?
What does the Euclidean Algorithm find for two numbers with one of them being zero?
What is the Euclidean Algorithm's time complexity in the average case?
let a,b ∈ *Z⁺*. then (a,b)[a,b] = ab
What is the complexity of the Euclidean Algorithm for finding the GCD of two numbers?
let a₁, a₂...aₙ ∈ *Z* and p prime ∈ *Z*. if p|a₁, a₂...aₙ, then p|aᵢ for some i
What is the GCD of 15 and 35 when using the Euclidean Algorithm?
Euclid geometry is the plane and solid geometry commonly taught in secondary schools.
What is the Euclidean Algorithm's primary limitation when dealing with very large numbers?
let n ∈ *Z*. then n is said to be even if 2|n and n is said to be odd if 2∤n
When was the Euclidean Algorithm first documented?
if a,b ∈ *Z*, a≥b>0, and a = bq + r with q,r ∈ *Z*, then (a,b) = (b,r)
What does the Euclidean Algorithm find for two numbers with one of them being negative?
What does the Euclidean Algorithm find for two numbers with one of them being a negative fraction?
let n be a composite number. then n has a prime divisor p with p≤√n
What is the step after calculating the remainder in the Euclidean Algorithm?
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