MATH 1350 NUMBER THEORY
Tests for Divisibility
Divisibility by Test
2
The number must be even
3
The sum of the digits must be divisible by 3
4
The last two digits must form a number divisible by 4
5
The last digit must be 0 or 5
6
The number must be divisible by 2 and 3
8
The last three numbers must form a number divisible by 8
9
The sum of the digits must be divisible by 9
10
The last digit must be 0
11
The difference of the sums of the digits in even and odd positions must be
divisible by 11
Divisibility by Products
If a natural number n is divisible by both a and b and a and b have no common factors other than
1, then n is also divisible by ab.
Greatest Common Divisor
The greatest common divisor of a set of natural numbers is the largest number that is a divisor
(factor) of all the numbers in the set. Symbol: GCD(m,n) = the greatest common divisor of m
and n.
Least Common Multiple
The least common multiple of a set of natural numbers is the smallest number that is a multiple
of all the numbers in the set. Symbol: LCM(m,n) = the least common multiple of m and n.
Venn Diagram Method for Finding GCD(m,n) and LCM(m,n)
• Find the prime factorization of each number. List all prime factors of m in one set and all
prime factors of n in another set accounting for multiplicity with as many common prime
factors as you can in the intersection.
• GCD(m,n) is the product of all prime factors in the intersection of the sets.
• LCM(m,n) is the product of all prime factors in the union of the sets.
Prime Factorization Method for Finding GCD(m,n) and LCM(m,n)
• Find the prime factorization of each number.
• To find the GCD, list all the prime factors (once) that appear in both factorizations.
Raise each factor to the lowest exponent to which it is raised in the two factorizations.
• To find the LCM, list all the prime factors (once) that appear in either factorization.
Raise each factor to the highest exponent to which it is raised in the two factorizations.