"A divisibility rule is a shorthand and useful way of determining whether a given integer is divisible by a fixed divisor without performing the division, usually by examining its digits."
Study of how numbers can be divided evenly without leaving a remainder.
Division and Factors: Understanding the basics of division and the factors of a number is essential to understanding divisibility.
Prime and Composite Numbers: Knowing the difference between prime and composite numbers is important in determining whether a number is divisible by another.
Divisibility Rules: There are various rules for determining divisibility by different numbers such as 2, 3, 4, 5, 6, 8, 9, and 10.
Greatest Common Divisor (GCD): The GCD is the largest number that divides two or more numbers.
Least Common Multiple (LCM): The LCM is the smallest number that is a multiple of two or more numbers.
Rational and Irrational Numbers: Understanding the properties of rational and irrational numbers can help to determine whether a number is divisible by another.
Euclid's Algorithm: This is a method for finding the GCD of two numbers through repeated division.
Modular Arithmetic: Modulo arithmetic involves finding the remainder of a division, and is useful in determining divisibility.
Sieve of Eratosthenes: This is a method for finding all prime numbers up to a certain limit.
Fundamental Theorem of Arithmetic: Every integer can be expressed uniquely as the product of prime factors.
Perfect Numbers: A perfect number is a number that is equal to the sum of its divisors, excluding itself.
Mersenne Primes: These are prime numbers of the form 2^n: Where n is a positive integer.
Fermat's Little Theorem: This theorem states that if p is a prime number and a is an integer, then a^(p-1) is congruent to 1 modulo p.
Wilson's Theorem: This theorem states that for any prime number p, (p-1)! is congruent to -1 modulo p.
Chinese Remainder Theorem: This theorem states that given a set of congruences with pairwise coprime moduli, there exists a unique solution modulo the product of the moduli.
Euler's Theorem: This theorem generalizes Fermat's Little Theorem to any positive integer, and states that if a and n are coprime, then a^phi(n) is congruent to 1 modulo n, where phi(n) is Euler's totient function.
Cryptography: Divisibility and number theory are fundamental to many cryptographic algorithms, such as RSA encryption.
Goldbach's Conjecture: This conjecture states that every even integer greater than 2 can be expressed as the sum of two prime numbers.
Twin Primes Conjecture: This conjecture states that there are infinitely many pairs of prime numbers that differ by 2.
Collatz Conjecture: This conjecture involves a simple sequence of integer operations, and is one of the most famous unsolved problems in mathematics.
Divisibility by 2: A number is divisible by 2 if it is an even number (i.e., it can be divided by 2 without leaving a remainder).
Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
Divisibility by 4: A number is divisible by 4 if its last two digits are divisible by 4.
Divisibility by 5: A number is divisible by 5 if its last digit is either 0 or 5.
Divisibility by 6: A number is divisible by 6 if it is divisible by both 2 and 3.
Divisibility by 7: There is no easy rule for testing divisibility by 7, but one can use the trick of repeated subtraction (i.e., divide the number by 7, subtract the quotient from the original number, and repeat the process until the remaining number is less than 7).
Divisibility by 8: A number is divisible by 8 if its last three digits are divisible by 8.
Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
Divisibility by 10: A number is divisible by 10 if its last digit is 0.
Divisibility by 11: A number is divisible by 11 if the difference between the sum of its digits in odd positions and the sum of its digits in even positions is divisible by 11.
Divisibility by 12: A number is divisible by 12 if it is divisible by both 3 and 4.
Divisibility by 13: There is no easy rule for testing divisibility by 13, but one can use the trick of repeated subtraction (i.e., divide the number by 13, subtract the quotient from the original number multiplied by 10, and repeat the process until the remaining number is less than 13).
Divisibility by 14: A number is divisible by 14 if it is divisible by both 2 and 7.
Divisibility by 15: A number is divisible by 15 if it is divisible by both 3 and 5.
Divisibility by 16: A number is divisible by 16 if its last four digits are divisible by 16.
Divisibility by 17: There is no easy rule for testing divisibility by 17, but one can use the trick of repeated subtraction (i.e., divide the number by 17, subtract the quotient from the original number multiplied by 10, and repeat the process until the remaining number is less than 17).
Divisibility by 18: A number is divisible by 18 if it is divisible by both 2 and 9.
Divisibility by 19: There is no easy rule for testing divisibility by 19, but one can use the trick of repeated subtraction (i.e., divide the number by 19, subtract the quotient from the original number multiplied by 10, and repeat the process until the remaining number is less than 19).
Divisibility by 20: A number is divisible by 20 if it is divisible by both 4 and 5.
"to determine whether a given integer is divisible by a fixed divisor without performing the division."
"decimal, or base 10, numbers."
"Martin Gardner."
"in his September 1962 'Mathematical Games' column in Scientific American."
"Mathematical Games."
"Yes, there are divisibility tests for numbers in any radix or base."
"No, they are all different."
"Its digits."
"No, a divisibility rule allows determining divisibility without performing the division."
"A shorthand and useful way to check if a number can be divided by another number without doing the actual division."
"It was where Martin Gardner explained and popularized the divisibility rules."
"although there are divisibility tests for numbers in any radix, or base, and they are all different, this article presents rules and examples only for decimal, or base 10, numbers."
"Yes, a divisibility rule can be used for any fixed divisor."
"No, the divisibility rule avoids the need for performing division."
"It provides a quicker way to determine divisibility without the actual division process."
"Yes, divisibility rules are primarily used in mathematical calculations."
"Yes, divisibility rules can be applied to integers of any size."
"Yes, this article presents rules and examples only for decimal, or base 10, numbers."
"The purpose of divisibility rules is to quickly determine if a given number is divisible by a fixed divisor without the need for actual division."