proof that $\tau (n)$ is the number of positive divisors of $n$
The following is a proof that $\tau $ counts the positive divisors^{} of its input (which must be a positive integer).
Proof.
Recall that $\tau $ behaves according to the following two rules:

1.
If $p$ is a prime and $k$ is a nonnegative integer, then $\tau ({p}^{k})=k+1$.

2.
If $\mathrm{gcd}(a,b)=1$, then $\tau (ab)=\tau (a)\tau (b)$.
Let $p$ be a prime. Then ${p}^{0}=1$. Note that 1 is the only positive divisor of 1 and $\tau (1)=\tau ({p}^{0})=0+1=1$.
Suppose that, for all positive integers $m$ smaller than $z\in \mathbb{Z}$ with $z>1$, the number of positive divisors of $m$ is $\tau (m)$. Since $z>1$, there exists a prime divisor^{} $p$ of $z$. Let $k$ be a positive integer such that ${p}^{k}$ exactly divides $z$. Let $a$ be a positive integer such that $z={p}^{k}a$. Then $\mathrm{gcd}(a,p)=1$. Thus, $\mathrm{gcd}(a,{p}^{k})=1$. Since $$, by the induction hypothesis, there are $\tau (a)$ positive divisors of $a$.
Let $d$ be a positive divisor of $z$. Let $y$ be a nonnegative integer such that ${p}^{y}$ exactly divides $d$. Thus, $0\le y\le x$, and there are $k+1$ choices for $y$. Let $c$ be a positive integer such that $d={p}^{y}c$. Then $\mathrm{gcd}(c,p)=1$. Since $c$ divides $d$ and $d$ divides $z$, we conclude that $c$ divides $z$. Since $c$ divides ${p}^{k}a$ and $\mathrm{gcd}(c,p)=1$, it must be the case that $c$ divides $a$. Thus, there are $\tau (a)$ choices for $c$. Since there are $k+1$ choices for $y$ and there are $\tau (a)$ choices for $c$, there are $(k+1)\tau (a)$ choices for $d$. Hence, there are $(k+1)\tau (a)$ positive divisors of $z$. Since $\tau (z)=\tau ({p}^{k}a)=\tau ({p}^{k})\tau (a)=(k+1)\tau (a)$, it follows that, for every positive integer $n$, the number of positive divisors of $n$ is $\tau (n)$. ∎
Title  proof that $\tau (n)$ is the number of positive divisors of $n$ 

Canonical name  ProofThattaunIsTheNumberOfPositiveDivisorsOfN 
Date of creation  20130322 13:30:24 
Last modified on  20130322 13:30:24 
Owner  Wkbj79 (1863) 
Last modified by  Wkbj79 (1863) 
Numerical id  11 
Author  Wkbj79 (1863) 
Entry type  Proof 
Classification  msc 11A25 