Divisor

A divisor of an integer $n$ is an integer $d$ satisfying the divisibility relation $d\mid n$. Equivalently, $n=dk$ for some integer $k$. The word positive matters when listing or counting divisors: unless signs are restricted, both $d$ and $-d$ qualify.

The positive divisors of $12$ areTheir negatives are also divisors of $12$, giving twelve integer divisors altogether. The same list of positive divisors works for $-12$. One has only the positive divisor $1$, while zero has every integer as a divisor, including zero under our divisibility convention. A nonzero integer never has zero as a divisor.

For a positive integer $n$, the positive divisors come in pairs $(d,n/d)$ whose product is $n$. For $12$, the pairs are $(1,12)$, $(2,6)$ and $(3,4)$. Each pair contains a divisor at most $\sqrt{n}$, so searching through candidate divisors up to the square root suffices to find all pairs. A pair repeats its own member only when $n$ is a perfect square; for $36$, the middle pair is $(6,6)$. Consequently a positive integer has an odd number of positive divisors exactly when it is a perfect square.

The prime factorization gives another complete description. Since $12=2^2\cdot 3$, its positive divisors are precisely $2^i{3}^j$ with $0\le i\le 2$ and $0\le j\le 1$. These independent choices give $(2+1)(1+1)=6$ positive divisors, the value of the divisor function $\tau (12)$. A prime factor is one particular kind of divisor; $4$, $6$ and $12$ show that divisors need not be prime numbers.

Thinking in pairs connects exact grouping with integer factorization: twelve objects can form three groups of four or four groups of three. Thinking in shared divisors gives the greatest common divisor; for $12$ and $18$, the positive common divisors are $1,2,3,6$, and the largest is $6$.