The Number Of Positive Divisors Of An Integer N D N

The Number of Positive Divisors of an Integer

In number theory, we denote the function d(n) to be the number of positive divisors of an integer n. For example, the integer 18 has positive divisors 1, 2, 3, 6, 9, and 18. Thus d(18) = 6, since there are strictly 6 positive divisors of 18. Thus:

(1)

\begin{align} d(n) = \sum_{d \mid n} 1 \end{align}

Lemma 1: If p is a prime, then d(p) = 2.

• Proof: By definition, a prime only has two positive divisors which are namely 1 and p. Thus d(p) = 2.

Lemma 2: If p is a prime, then d(p²) = 3.

• Proof: By definition, a prime only has two positive divisors which are namely 1 and p. Hence 1 | p², and p | p², but also p² | p². Thus d(p²) = 3.

Lemma 3: If p is a prime, then for e = 1, 2, 3, … d(p^e) = e + 1.

• Proof: Suppose that we have the prime p raised to the e^th power. It thus follows that the divisors of p^e are 1, p, p², p³, …, p^e-1, p^e. So in total, there are e + 1 positive factors of p^e, so d(p^e) = e + 1

Lemma 4: If p and q are prime, then d(pq) = 4.

• Proof: Since p and q are prime, the 1 | pq, p | pq, q | pq, and of course pq | pq. Thus there are exactly 4 positive divisors of pq, and thus d(pq) = 4.

Lemma 5: If p₁^e₁p₂^e₂p₃^e₃…p_k^e_k is the prime power decompositions of n, then d(n) = d(p₁^e₁)d(p₂^e₂)d(p₃^e₃)…d(p_k^e_k) for e_k ≥ 1