"A divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n."
Factors and multiples that are shared by two or more numbers.
Factors and Multiples: A factor of a positive integer is any number that divides it without leaving a remainder. A multiple of a positive integer is any number that is obtained by multiplying it by another positive integer.
Prime Numbers: A prime number is a positive integer greater than 1 that has no positive integer divisors other than 1 and itself.
Composite Numbers: A composite number is a positive integer that is not prime, i.e., it has two or more positive integer divisors.
Greatest Common Factor (GCF): The GCF of two or more positive integers is the largest positive integer that divides each of them without leaving a remainder.
Least Common Multiple (LCM): The LCM of two or more positive integers is the smallest positive integer that is a multiple of each of them.
Prime Factorization: Prime factorization is the process of decomposing a positive integer into a product of its prime factors.
Euclidean Algorithm: The Euclidean algorithm is an efficient algorithm for finding the GCF of two positive integers.
Sieve of Eratosthenes: The Sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to a given limit.
Division Algorithm: The division algorithm is a procedure for dividing one positive integer by another, producing both a quotient and a remainder.
Common Factor and Multiple: A common factor of two or more positive integers is any positive integer that divides each of them without leaving a remainder, whereas a common multiple of two or more positive integers is any positive integer that is a multiple of each of them.
Arithmetic Sequences: An arithmetic sequence is a sequence of numbers where each term is obtained by adding a fixed amount to the preceding term.
Geometric Sequences: A geometric sequence is a sequence of numbers where each term is obtained by multiplying the preceding term by a fixed amount.
Diophantine Equations: Diophantine equations are equations in which only integer solutions are sought.
Bezout's Identity: Bezout's identity is a theorem in number theory stating that for any two positive integers a and b, there exist integers m and n such that am + bn = gcd(a, b).
Euclid's Lemma: Euclid's lemma is a theorem in number theory stating that if a prime number divides the product of two positive integers, then it must divide at least one of the integers.
Chinese Remainder Theorem: The Chinese Remainder Theorem is a theorem in number theory that describes the solutions of a system of linear congruences with pairwise coprime moduli.
Modular Arithmetic: Modular arithmetic is a system of arithmetic in which numbers "wrap around" after reaching a certain value called the modulus.
Fermat's Little Theorem: Fermat's Little Theorem is a theorem in number theory stating that if p is a prime number and a is not divisible by p, then a^(p-1) is congruent to 1 modulo p.
Wilson's Theorem: Wilson's Theorem is a theorem in number theory stating that a positive integer n is prime if and only if (n-1)! + 1 is divisible by n.
Prime Factors: The prime numbers that divide two or more numbers are called common prime factors. For example, the common prime factors of 12 and 20 are 2.
Greatest Common Factor (GCF): The largest common factor of two or more numbers is called the greatest common factor. For example, the GCF of 20 and 30 is 10.
Composite Factors: The composite numbers that divide two or more numbers are called common composite factors. For example, the common composite factors of 8 and 12 are 2 and 4.
Odd Factors: The odd factors that divide two or more odd numbers are called common odd factors. For example, the common odd factors of 15 and 21 are 3.
Even Factors: The even factors that divide two or more even numbers are called common even factors. For example, the common even factors of 8 and 20 are 2 and 4.
Square Factors: The square numbers that divide two or more numbers are called common square factors. For example, the common square factors of 16 and 36 are 4.
Least Common Multiple (LCM): The smallest common multiple of two or more numbers is called the least common multiple. For example, the LCM of 8 and 12 is 24.
Prime Multiples: The multiples of the prime numbers that are common to two or more numbers are called common prime multiples. For example, the common prime multiples of 3 and 5 are 15, 30, 45, etc.
Composite Multiples: The multiples of composite numbers that are common to two or more numbers are called common composite multiples. For example, the common composite multiples of 4 and 6 are 12, 24, 36, etc.
Even Multiples: The even multiples that are common to two or more numbers are called common even multiples. For example, the common even multiples of 6 and 8 are 24, 48, 72, etc.
Odd Multiples: The odd multiples that are common to two or more odd numbers are called common odd multiples. For example, the common odd multiples of 3 and 5 are 15, 45, 75, etc.
Square Multiples: The multiples of square numbers that are common to two or more numbers are called common square multiples. For example, the common square multiples of 4 and 9 are 36, 144, 324, etc.
"In this case, one also says that n is a multiple of m."
"An integer n is divisible or evenly divisible by another integer m if m is a divisor of n."
"This implies dividing n by m leaves no remainder."
"In this case, one also says that n is a multiple of m."
"A factor of n."
"No, a divisor must be an integer."
"For example, 6 is a multiple of 2 since 6 = 2 × 3."
"If a number only has divisors 1 and itself, it is called a prime number."
"For example, 7 is a prime number as its only divisors are 1 and 7."
"Yes, every integer is divisible by 1."
"Yes, every integer is divisible by itself."
"For example, 12 is divisible by 2 since 12 = 2 × 6."
"We determine that a number is divisible by 3 if the sum of its digits is divisible by 3."
"For example, 15 is divisible by 5 since 15 = 5 × 3."
"The remainder will be zero if the two numbers have a perfect division."
"A number n is evenly divisible by another number m if dividing n by m gives a whole number result without any remainder."
"No, some numbers are not divisible by certain integers and will leave a remainder when divided."
"For example, 10 is not divisible by 3 as 10 ÷ 3 leaves a remainder of 1."
"A factor of an integer n is another number that can be multiplied by some integer to produce n, which is equivalent to being a divisor of n."