Multiples and divisors are the most essential concepts in number theory and come up in nearly every problem.
Multiples
It seems natural to start with some definitions
A multiple of an integer is the product of said integer with any other integer
Given the number 4, we have multiples \dots,-4,0,4,8,\dots
The greatest common divisor between two integers m and n is notated
\gcd(m,n)
Divisors
A divisor or factor of an integer n is any other integer d such that d\mid n
Note that every integer n has divisors 1 and n.
The least common multiple between two integers n and m is notated
\text{lcm}(m,n)
Properties
Total factors
Say you want to find the total number of factors for an integer. Instead of hashing it out manually, there exist an easier method.
Find the total number of factors to the integer 72.
We begin by finding the prime factorization of 72, in this case
72 = 2^3 * 3^2.
Notice that each factor of 72 contains between zero and three 2’s, and between zero and two 3’s. This gives us a total of 4 \times 3 = 12 factors for 72.
We use this trick to quickly calculate the total number of factors for an integer!
The total factors of an integer
n = p_1^{k_1} \cdot p_2^{k_2} \cdots p_m^{k_m}
is the product (k_1 + 1)(k_2 + 1)\cdots(k_m + 1)
Problems
Essential
How many positive and negative integers are factors of 12?
The number 12 can be factored 2^2*3^1. This gives a total of 6 factors. Including negative factors gives a total of 12 integers.