"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."
Numbers that can be evenly divided into another number, and numbers that are products of a given number and another whole number.
Factors: A factor is a number that can divide another number without leaving any remainder.
Multiples: A multiple is a number that is the product of some other number and an integer.
Common factors: When two or more numbers have a factor in common, it is called a common factor.
Common multiples: When two or more numbers have a multiple in common, it is called a common multiple.
Prime factorization: Prime factorization is the process of finding the prime factors of a number.
Greatest common factor: The greatest common factor of two or more numbers is the highest common factor that divides all of the numbers.
Least common multiple: The least common multiple of two or more numbers is the smallest number that is a multiple of all of the numbers.
Divisibility rules: Divisibility rules are shortcuts that can help you quickly determine if a number is divisible by another number.
Properties of factors and multiples: There are several properties of factors and multiples that can help you solve problems more quickly and easily.
Applications of factors and multiples: Factors and multiples are used in many real-world applications, such as finding the common denominator of fractions, calculating the LCM and GCF of multiple numbers, and factoring polynomials.
Prime numbers: Numbers that are divisible only by 1 and themselves. For example, 2, 3, 5, 7, 11, and 13 are all prime numbers.
Composite numbers: Numbers that are not prime and can be factored into smaller numbers. For example, 6, 8, 10, and 12 are all composite numbers, since they can be factored into smaller numbers (e.g. 6 = 2 x 3, 8 = 2 x 2 x 2, etc.).
Even numbers: Numbers that are divisible by 2. For example, 2, 4, 6, 8, and 10 are all even numbers.
Odd numbers: Numbers that are not divisible by 2. For example, 1, 3, 5, 7, and 9 are all odd numbers.
Square numbers: Numbers that are the square of a whole number. For example, 4, 9, 16, and 25 are all square numbers, since they are the squares of 2, 3, 4, and 5, respectively.
Cube numbers: Numbers that are the cube of a whole number. For example, 8, 27, and 64 are all cube numbers, since they are the cubes of 2, 3, and 4, respectively.
Perfect numbers: Numbers that are the sum of their proper divisors (i.e. all of their divisors except for themselves). For example, 6 is a perfect number, since its proper divisors are 1, 2, and 3, and 1 + 2 + 3 = 6.
Abundant numbers: Numbers that are greater than the sum of their proper divisors. For example, 12 is an abundant number, since its proper divisors are 1, 2, 3, 4, and 6, and 1 + 2 + 3 + 4 + 6 = 16, which is less than 12.
Deficient numbers: Numbers that are less than the sum of their proper divisors. For example, 8 is a deficient number, since its proper divisors are 1, 2, and 4, and 1 + 2 + 4 = 7, which is less than 8.
Twin primes: A pair of prime numbers that differ by 2. For example, (3,5), (5,7), and (11,13) are all twin primes.
Highly composite numbers: Numbers with more divisors than any smaller number. For example, 12 is a highly composite number, since it has 6 divisors (1, 2, 3, 4, 6, and 12), which is more than any smaller number.
Amicable numbers: Two numbers are amicable if the sum of the proper divisors of one equals the other number, and vice versa. For example, (220, 284) is an amicable pair, since the proper divisors of 220 add up to 284 (1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284), and the proper divisors of 284 add up to 220 (1 + 2 + 4 + 71 + 142 = 220).
Semiprimes: Numbers that are the product of two prime numbers. For example, 15 is a semiprime, since it is the product of 3 and 5, both of which are prime.
Square-free numbers: Numbers that are not divisible by any perfect square (other than 1). For example, 10, 14, and 15 are all square-free numbers, since they are not divisible by any perfect square other than 1.
"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."