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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.

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

**corollary 1.20 (1.5, pg. 30)**

let a₁, a₂...aₙ ∈ *Z* and p prime ∈ *Z*. if p|a₁, a₂...aₙ, then p|aᵢ for some i

**corollary 1.15 (1.5, pg. 26)**

let n ∈ *Z*. then n is said to be even if 2|n and n is said to be odd if 2∤n

**even/odd definition (1.1, pg. 8)**

What is the final result of the Euclidean Algorithm for 14 and 9?

**1. GCD is 1**- 2. GCD is 2
- 3. GCD is 3
- 4. GCD is 4

Who is the founder of the Euclid geometry?

- Spanish mathematician; Euclid
**Greek mathematician; Euclid..**- Italian mathematician; Euclid
- None of the above

What does the Euclidean Algorithm find for two numbers with one of them being a decimal?

- 1. GCD is the whole number part
**2. GCD is 1**- 3. GCD is the decimal part
- 4. GCD is 2

What is the output of the Euclidean Algorithm for 17 and 11?

- 1. GCD is 1
- 2. GCD is 2
**3. GCD is 1 (unchanged)**- 4. GCD is 3

What is the result of applying the Euclidean Algorithm to 99 and 33?

- 1. GCD is 3
**2. GCD is 33**- 3. GCD is 9
- 4. GCD is 11

there are infinitely many primes numbers p in which p+2 is also a prime number

**twin prime conjecture (1.2, pg. 14)**

Euclidean geometry is an axiomatic system, in which all theorems are derived from a small number of axioms.

**True**- False
- Indecisive
- No Idea

What is the Euclidean Algorithm's worst-case time complexity when both numbers are equal?

- 1. O(n)
- 2. O(n log n)
- 3. O(log n)
**4. O(1)**

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

**proposition 1.2 (1.1, pg. 5)**

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.

**prime number sieve (pg.12)**

What does the Euclidean Algorithm find for coprime numbers?

**1. GCD is 1**- 2. GCD is 2
- 3. GCD is 3
- 4. GCD is 4

let a,b,p ∈ *Z* with p being prime. if p|ab, then p|a or p|b

**lemma 1.14 (related to euclid's theorem) (1.5, pg. 26)**

What is the primary purpose of the Euclidean Algorithm in computer science?

**1. Optimization of algorithms**- 2. Sorting arrays
- 3. Data storage
- 4. Graph theory

What is the GCD of 63 and 35 when using the Euclidean Algorithm?

- 1. GCD is 3
**2. GCD is 7**- 3. GCD is 5
- 4. GCD is 9

What is the first step in each iteration of the Euclidean Algorithm?

**1. Divide the divisor by the remainder**- 2. Multiply the divisor by the remainder
- 3. Subtract the divisor from the remainder
- 4. Add the divisor to the remainder

Which step follows calculating the quotient in the Euclidean Algorithm?

- 1. Swap the numbers
- 2. Subtract the numbers
- 3. Repeat the algorithm
**4. Divide the new divisor by the new remainder**

*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

**pairwise relatively prime definition (1.3, pg. 20)**

What is the common notation for denoting the GCD of two numbers using the Euclidean Algorithm?

- 1. GCD(a, b)
**2. (a, b)**- 3. GCM[a, b]
- 4. GC[a, b]

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

**greatest integer function definition (1.1, pg. 6)**

let a,b ∈ *Z* with (a,b) = d. then (a/d, b/d) = 1

**proposition 1.10 (1.3, pg. 19)**

see example in book

**the euclidian algorithm theorem (1.4, pg. 23)**

What does the Euclidean Algorithm find for two prime numbers?

**1.**- GCD is 1
- 2. GCD is 2
- 3. GCD is 3
- 4. GCD is the larger prime number

Which step is repeated in the Euclidean Algorithm until a GCD is found?

- 1. Multiplication
**2. Subtraction**- 3. Addition
- 4. Division

What is the first step in the Euclidean Algorithm?

**1. Divide the larger number by the smaller number**- 2. Subtract the numbers
- 3. Swap the numbers
- 4. Multiply the numbers

What is the starting point of the Euclidean Algorithm?

- 1. Addition of the numbers
- 2. Subtraction of the numbers
**3. Division of the numbers**- 4. Multiplication of the numbers

What is the Euclidean Algorithm's primary application in computer algorithms?

**1. Optimizing recursive algorithms**- 2. Sorting algorithms
- 3. Searching algorithms
- 4. Graph algorithms

What is the output of the Euclidean Algorithm for 12 and 7?

- 1. GCD is 1
- 2. GCD is 2
**3. GCD is 1 (unchanged)**- 4. GCD is 3

What is the main purpose of the Euclidean Algorithm in number theory?

- 1. Finding prime numbers
- 2. Finding composite numbers
**3. Calculating the greatest common divisor**- 4. Calculating the least common multiple

When using the Euclidean Algorithm, what happens when the remainder becomes zero?

- 1. Stop the algorithm
**2. The divisor is the GCD**- 3. The quotient is the GCD
- 4. Continue with another iteration

In which branch of mathematics is the Euclidean Algorithm extensively used?

- 1. Geometry
**2. Number theory**- 3. Calculus
- 4. Algebra

What is the key idea in the Euclidean Algorithm for finding the GCD?

- 1. Multiplying the numbers
- 2. Adding the numbers
- 3. Exponentiation
**4. Repeated subtraction and division**

What is the output of the Euclidean Algorithm for 77 and 55?

- 1. GCD is 1
- 2. GCD is 2
- 3. GCD is 3
**4. GCD is 11**

What is the GCD of 72 and 90 when using the Euclidean Algorithm?

- 1. GCD is 6
- 2. GCD is 8
- 3. GCD is 10
**4. GCD is 18**

What property of numbers does the Euclidean Algorithm utilize to find the GCD?

- 1. Associativity
- 2. Distributivity
**3. Transitivity**- 4. Commutativity

What is the GCD of 33 and 14 when using the Euclidean Algorithm?

- 1. GCD is 1
- 2. GCD is 2
- 3. GCD is 3
**4. GCD is 11**

What is the alternative name for the Euclidean Algorithm?

- 1. Pythagorean Algorithm
**2. Division Algorithm**- 3. Factorization Algorithm
- 4. Quadratic Algorithm

What does the Euclidean Algorithm find for two numbers with one of them being a negative fraction?

- 1. GCD is the numerator
- 2. GCD is the denominator
- 3. GCD is 1
**4. GCD is 1 (unchanged)**

How many dimensions does a surface have?

- 1
**2**- 3
- 4

What is the Euclidean Algorithm's main advantage over other GCD calculation methods?

- 1. Speed
- 2. Simplicity
**3. Efficiency for large numbers**- 4. Accuracy

Which of the following is NOT a step in the Euclidean Algorithm?

- 1. Divide the larger number by the smaller number
- 2. Subtract the remainder from the divisor
- 3. Swap the positions of the numbers
**4. Add the numbers together**

let a, b ∈ *Z* with b > 0. then there exists unique q, r ∈ *Z* such that a = bq + r and 0≤r<b

**the division algorithm (1.1, pg. 7)**

What is the Euclidean Algorithm's main limitation when dealing with floating-point numbers?

**1. Precision loss**- 2. Computation time
- 3. Memory usage
- 4. Complexity

Which operation is at the core of the Euclidean Algorithm?

- 1. Multiplication
**2. Subtraction**- 3. Addition
- 4. Division

lim π(x)lnx = 1x→∞ (x)

**prime number theorem (1.2, pg. 14)**

What is the result of applying the Euclidean Algorithm to 48 and 18?

**1. GCD is 6**- 2. GCD is 8
- 3. GCD is 12
- 4. GCD is 10

How many dimensions does a solid have?

- 1
- 2
**3**- 4

What is the GCD of 15 and 35 when using the Euclidean Algorithm?

- 1. GCD is 1
**2. GCD is 5**- 3. GCD is 7
- 4. GCD is 9

Which ancient civilization is closely associated with the early use of the Euclidean Algorithm?

- 1. Babylonian civilization
**2. Ancient Greek civilization**- 3. Egyptian civilization
- 4. Chinese civilization

What is the main advantage of the Euclidean Algorithm in terms of memory usage?

**1. It requires minimal memory**- 2. It uses no memory
- 3. It uses a constant amount of memory
- 4. It uses a large amount of memory

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

**least common multiple definition (1.5, pg. 28)**

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

**prime number definition (1.2, pg. 10)**

What does the Euclidean Algorithm find for two numbers with one of them being a fraction?

- 1. GCD is the numerator
- 2. GCD is the denominator
**3. GCD is 1**- 4. GCD is 2

Which ancient mathematician is believed to have used a precursor to the Euclidean Algorithm?

- 1. Archimedes
- 2. Pythagoras
**3. Euclid of Alexandria**- 4. Aristotle

The boundaries of solids are called?

- Curves
- Points
**Surfaces**- Lines

every even integer greater than 2 can be expressed as the sum of 2 (not necessarily distinct) primes

**goldbach's conjecture (1.2, pg. 15)**

let n be a composite number. then n has a prime divisor p with p≤√n

**proposition 1.7 (1.2, pg. 11)**

What is the GCD of 42 and 56 when using the Euclidean Algorithm?

- 1. GCD is 4
- 2. GCD is 7
**3. GCD is 14**- 4. GCD is 21

How many dimensions does a point have?

- 0
**1**- 2
- 3

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

**greatest common divisor definition (1.3, pg. 18)**

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)

**proposition 1.8 (1.2, pg. 13)**

What is the Euclidean Algorithm's primary limitation when dealing with very large numbers?

- 1. Inaccuracy
**2. Computation time**- 3. Complexity
- 4. Memory usage

there are infinitely many prime numbers*understanding this proof helps with other proofs

**euclid's theorem (1.2, pg. 11)**

What is the main goal of the Euclidean Algorithm?

- 1. Finding the smallest number
**2. Finding the largest common factor**- 3. Finding the average of the numbers
- 4. Finding the product of the numbers

What is the step after calculating the remainder in the Euclidean Algorithm?

- 1. Swap the numbers
**2. Repeat with the divisor as the new divisor and the remainder as the new dividend**- 3. Add the divisor and remainder
- 4. Multiply the divisor and remainder

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

**fermant prime (1.2, pg. 16)**

keep in mind that the gcd of the two numbers you start with is the remainder that happens before the remainder hits 0

**euclidian algorithm**

What shape are side faces of a pyramid?

**Triangle**- Square
- Polygons
- Trapeziums

What is the output of the Euclidean Algorithm for 20 and 8?

- 1. GCD is 2
**2. GCD is 4**- 3. GCD is 6
- 4. GCD is 8

What is the final step in each iteration of the Euclidean Algorithm?

- 1. Subtract the remainder from the divisor
- 2. Multiply the remainder by the divisor
**3. Swap the numbers**- 4. Divide the divisor by the remainder

What is the primary application of the Euclidean Algorithm in cryptography?

- 1. Public key generation
- 2. Data encryption
- 3. Hashing
**4. Integer factorization**

What does the Euclidean Algorithm find for two numbers with one of them being negative?

- 1. GCD is the positive number
- 2. GCD is the negative number
- 3. GCD is 1
**4. GCD is the positive number**

What is the next step after swapping the numbers in the Euclidean Algorithm?

- 1. Repeat the algorithm
**2. Divide the new divisor by the new remainder**- 3. Subtract the numbers
- 4. Multiply the numbers

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

**fundamental theorem of arithmetic (unique prime factorization) (1.5, pg. 27)**

What is the last step in the Euclidean Algorithm?

- 1. Divide the remainder by the divisor
- 2. Multiply the remainder by the divisor
**3. The divisor is the GCD**- 4. Repeat the algorithm with a new divisor and remainder

What is the GCD of 36 and 48 when using the Euclidean Algorithm?

- 1. GCD is 4
**2. GCD is 12**- 3. GCD is 8
- 4. GCD is 6

let a,b ∈ *Z⁺*. then (a,b)[a,b] = ab

**theorem 1.19 (1.5, pg. 30)**

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

**direchlet's theorem on prime numbers (1.5, pg. 31)**

What is the fundamental principle behind the Euclidean Algorithm?

- 1. Pythagorean theorem
- 2. Calculus
**3. Repeated subtraction and division**- 4. Quadratic formula

let a, b ∈ *Z*. if a|b and b|c, then a|c

**proposition 1.1 (1.1, pg. 5)**

What is the Euclidean Algorithm's time complexity in the average case?

- 1. O(n)
- 2. O(n log n)
**3. O(log n)**- 4. O(n^2)

What does the Euclidean Algorithm find for two identical numbers?

- 1. GCD is 1
- 2. GCD is 2
- 3. GCD is 3
**4. GCD is the number itself**

The three steps from solids to points are?

**Solids – Surfaces – Lines – Points**- Solids – Lines – Surfaces- Points
- Lines – Surfaces – Solids – Points
- None of the above.

n is a perfect square iff the exponents in the prime factors are all even

**theorem from class**

Hyperbolic geometry means the same thing as Euclidean geometry.

- True
**False**- Indecisive
- No Idea

What is the GCD of 56 and 63 when using the Euclidean Algorithm?

- 1. GCD is 3
**2. GCD is 7**- 3. GCD is 5
- 4. GCD is 9

Euclid geometry is the plane and solid geometry commonly taught in secondary schools.

**True**- False
- Indecisive
- No Idea

What is the next step after calculating the quotient in the Euclidean Algorithm?

- 1. Swap the numbers
- 2. Subtract the numbers
**3. Repeat the algorithm with a new divisor and remainder**- 4. Divide the remainder by the divisor

What is the Euclidean Algorithm's space complexity?

- 1. O(n)
- 2. O(n log n)
**3. O(1)**- 4. O(log n)

What is the worst-case time complexity of the Euclidean Algorithm?

- 1. O(n)
- 2. O(n log n)
**3. O(log n)**- 4. O(n^2)

What is the result of using the Euclidean Algorithm for 56 and 48?

- 1. GCD is 4
**2. GCD is 8**- 3. GCD is 12
- 4. GCD is 16

What is the GCD of 9 and 10 when using the Euclidean Algorithm?

- 1. GCD is 1
- 2. GCD is 2
- 3. GCD is 3
**4. GCD is 1 (unchanged)**

What is the complexity of the Euclidean Algorithm for finding the GCD of two numbers?

- 1. Linear time
- 2. Quadratic time
**3. Logarithmic time**- 4. Exponential time

every integer greater than 1 has a prime divisor

**lemma 1.5 (1.2, pg. 11)**

What is the GCD of 60 and 84 when using the Euclidean Algorithm?

- 1. GCD is 4
- 2. GCD is 6
- 3. GCD is 10
**4. GCD is 12**

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

**estimating the number of primes definition (1.2, pg. 14)**

any prime number expressible in the for 2ᵖ - 1 where p is prime is a mersenne primeex. 3 = 2²-1, 7, 31, 127, 8191

**mersenne prime (1.2, pg. 16)**

let a, b ∈ *Z*. a|b if there exists a c ∈*Z* such that b = ac

**divisibility definition (1.1, pg. 4)**

What is the Euclidean Algorithm's main advantage in terms of computational efficiency?

- 1. It is always faster than other methods
- 2. It guarantees the lowest GCD
**3. It runs in logarithmic time**- 4. It requires less memory

Which pair of numbers will result in a GCD of 1 when using the Euclidean Algorithm?

- 1. 30 and 15
- 2. 21 and 7
- 3. 24 and 12
**4. 25 and 8**

if a,b ∈ *Z*, a≥b>0, and a = bq + r with q,r ∈ *Z*, then (a,b) = (b,r)

**lemma 1.12 (1.4, pg. 23)**

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

**proposition 1.11 (1.3, pg. 19)**

What does the Euclidean Algorithm find for two numbers with one of them being zero?

- 1. GCD is 0
**2. GCD is the non-zero number**- 3. GCD is 1
- 4. GCD is 2

Who is the mathematician credited with the development of the Euclidean Algorithm?

- 1. Pythagoras
- 2. Euclid
**3. Euclid of Alexandria**- 4. Archimedes

When was the Euclidean Algorithm first documented?

- 1. 3rd century BC
**2. Circa 300 BC**- 3. 5th century AD
- 4. 10th century AD

What is the Euclidean Algorithm's space complexity for finding the GCD of two numbers?

- 1. O(n)
- 2. O(n log n)
**3. O(1)**- 4. O(log n)

What does the Euclidean Algorithm compute for two given numbers?

- 1. Least common multiple (LCM)
**2. Greatest common divisor (GCD)**- 3. Average of the numbers
- 4. Sum of the numbers

=> 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)

**proof tecniques**

What is the Euclidean Algorithm used for?

- 1. Solving linear equations
**2. Finding the greatest common divisor (GCD) of two integers**- 3. Calculating square roots
- 4. Factoring polynomials

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