Divisibility Rules Quiz

Divisibility Rules for Numbers 1-30

Divisibility rules are a shorthand way of determining whether a number is divisible by a fixed divisor without performing the division.
Martin Gardner popularized these rules in his 1962 "Mathematical Games" column in Scientific American.
The rules presented in this article are for decimal, or base 10, numbers.
To test for divisibility by a power of 2 or 5, only the last n digits need to be examined.
To test for divisibility by any number expressed as the product of prime factors, separately test for divisibility by each prime to its appropriate power.
The divisibility rules for numbers 1-30 transform a given number into a generally smaller number while preserving divisibility by the divisor of interest.
Divisibility by 2 can be determined by examining the last digit of the number.
Divisibility by 3 or 9 can be determined by adding together each digit in the number and checking if the sum is divisible by 3 or 9.
Divisibility by 4 can be determined by examining the last two digits of the number and checking if they are divisible by 4.
Divisibility by 5 can be determined by examining the last digit of the number and checking if it is 0 or 5.
Divisibility by 6 can be determined by checking if the number is both even and divisible by 3.
Divisibility by 7 can be tested by a recursive method, multiplication by the Ekhādika, or the Pohlman-Mass method.
These rules can be useful in math competitions or when quick calculations are needed.Divisibility Rules for Numbers
Divisibility by 2: A number is divisible by 2 if its last digit is even.
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 the last two digits of the number are divisible by 4.
Divisibility by 5: A number is divisible by 5 if its last digit is 0 or 5.
Divisibility by 6: A number is divisible by 6 if it is divisible by both 2 and 3.
Divisibility by 7: Various methods can be used to test divisibility by 7, including the Pohlman-Mass method, multiplication by 3 method, digit pair method, and positive sequence method.
Divisibility by 13: A remainder test can be used, involving multiplying the rightmost digit by a sequence of numbers and adding the results.
Divisibility by composite divisors: A number is divisible by a composite divisor if it is divisible by the highest power of each of its prime factors.
Divisibility by prime divisors: An inverse to 10 modulo the prime divisor can be found and used to create a rule for divisibility.
Generalized divisibility rule: To test for divisibility by a number ending in 1, 3, 7, or 9, a multiple of that number ending in 9 can be found and a formula using the multiple can be used to test for divisibility.
Proofs: Divisibility rules can be proven using algebraic manipulation, such as writing a number as the sum of each digit times a power of 10.
Divisibility beyond 30: Divisibility properties of numbers can be determined by finding the highest power of each prime factor and using the corresponding rule.
Notable examples: Rules for some more notable divisors, including 31, 37, 41, and 101, can be found using the prime divisor method.Rules for Divisibility by 2, 5, and 7
Divisibility by 2 depends on whether the last digit of a number is even or odd.
Divisibility by 5 depends on whether the last digit of a number is 0 or 5.
Divisibility by 10 depends on whether the last digit of a number is 0.
Divisibility by a factor of a power of 10 only depends on the last digit(s) of a number.
The rule "double the number formed by all but the last two digits, then add the last two digits" can be used for divisibility by 7 when only the last digit(s) are removed.
The representation of a number may be multiplied by any number relatively prime to the divisor without changing its divisibility.
The rule "subtract twice the last digit from the rest" can be used for divisibility by 7 when the last digit(s) is multiplied by a factor.
All the rules for divisibility by 2, 5, and 7 can be derived using modular arithmetic.
For divisibility by 2n or 5n, only the last n digits need to be checked.
For divisibility by 7, the rule "y − 2z is divisible by 7" can be used, where x is represented as 10 ⋅ y + z.
Divisibility by other factors can be proven using the basic fact that 10 mod m is invertible if 10 and m are relatively prime.
The rules for divisibility by 2, 5, and 7 are useful in mathematics and everyday life, such as in checking if a number is evenly divisible by a certain amount.

Divisibility Rules for Numbers 1-30

Divisibility rules are a shorthand way of determining whether a number is divisible by a fixed divisor without performing the division.
Martin Gardner popularized these rules in his 1962 "Mathematical Games" column in Scientific American.
The rules presented in this article are for decimal, or base 10, numbers.
To test for divisibility by a power of 2 or 5, only the last n digits need to be examined.
To test for divisibility by any number expressed as the product of prime factors, separately test for divisibility by each prime to its appropriate power.
The divisibility rules for numbers 1-30 transform a given number into a generally smaller number while preserving divisibility by the divisor of interest.
Divisibility by 2 can be determined by examining the last digit of the number.
Divisibility by 3 or 9 can be determined by adding together each digit in the number and checking if the sum is divisible by 3 or 9.
Divisibility by 4 can be determined by examining the last two digits of the number and checking if they are divisible by 4.
Divisibility by 5 can be determined by examining the last digit of the number and checking if it is 0 or 5.
Divisibility by 6 can be determined by checking if the number is both even and divisible by 3.
Divisibility by 7 can be tested by a recursive method, multiplication by the Ekhādika, or the Pohlman-Mass method.
These rules can be useful in math competitions or when quick calculations are needed.Divisibility Rules for Numbers
Divisibility by 2: A number is divisible by 2 if its last digit is even.
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 the last two digits of the number are divisible by 4.
Divisibility by 5: A number is divisible by 5 if its last digit is 0 or 5.
Divisibility by 6: A number is divisible by 6 if it is divisible by both 2 and 3.
Divisibility by 7: Various methods can be used to test divisibility by 7, including the Pohlman-Mass method, multiplication by 3 method, digit pair method, and positive sequence method.
Divisibility by 13: A remainder test can be used, involving multiplying the rightmost digit by a sequence of numbers and adding the results.
Divisibility by composite divisors: A number is divisible by a composite divisor if it is divisible by the highest power of each of its prime factors.
Divisibility by prime divisors: An inverse to 10 modulo the prime divisor can be found and used to create a rule for divisibility.
Generalized divisibility rule: To test for divisibility by a number ending in 1, 3, 7, or 9, a multiple of that number ending in 9 can be found and a formula using the multiple can be used to test for divisibility.
Proofs: Divisibility rules can be proven using algebraic manipulation, such as writing a number as the sum of each digit times a power of 10.
Divisibility beyond 30: Divisibility properties of numbers can be determined by finding the highest power of each prime factor and using the corresponding rule.
Notable examples: Rules for some more notable divisors, including 31, 37, 41, and 101, can be found using the prime divisor method.Rules for Divisibility by 2, 5, and 7
Divisibility by 2 depends on whether the last digit of a number is even or odd.
Divisibility by 5 depends on whether the last digit of a number is 0 or 5.
Divisibility by 10 depends on whether the last digit of a number is 0.
Divisibility by a factor of a power of 10 only depends on the last digit(s) of a number.
The rule "double the number formed by all but the last two digits, then add the last two digits" can be used for divisibility by 7 when only the last digit(s) are removed.
The representation of a number may be multiplied by any number relatively prime to the divisor without changing its divisibility.
The rule "subtract twice the last digit from the rest" can be used for divisibility by 7 when the last digit(s) is multiplied by a factor.
All the rules for divisibility by 2, 5, and 7 can be derived using modular arithmetic.
For divisibility by 2n or 5n, only the last n digits need to be checked.
For divisibility by 7, the rule "y − 2z is divisible by 7" can be used, where x is represented as 10 ⋅ y + z.
Divisibility by other factors can be proven using the basic fact that 10 mod m is invertible if 10 and m are relatively prime.
The rules for divisibility by 2, 5, and 7 are useful in mathematics and everyday life, such as in checking if a number is evenly divisible by a certain amount.