Suppose that $m=500$ and $1 \leq n \leq 499$ and $1 \leq r \leq 15$ and $2 \leq s \leq 9$ and $t=0$. Since $s>0$, then the algorithm says that $t$ is the remainder when $r$ is divided by $s$. Since $t=0$, then $r$ is a multiple of $s$. Thus, $r=a s$ for some positive integer $a$. Since $r>0$, then the algorithm says that $s$ is the remainder when $n$ is divided by $r$. In other words, $n=b r+s$ for some positive integer $b$. But $r=a s$, so $n=b a s+s=(b a+1) s$. In other words, $n$ is a multiple of $s$, say $n=c s$ for some positive integer $c$. Since $n>0$, then $r$ is the remainder when $m$ is divided by $n$. In other words, $m=d n+r$ for some positive integer $d$. But $r=a s$ and $n=c s$ so $m=d c s+a s=(d c+a) s$. In other words, $m$ is a multiple of $s$, say $m=e s$ for some positive integer $e$. But $m=500$ and $2 \leq s \leq 9$. Since $m$ is a multiple of $s$, then $s$ is a divisor of 500 and so the possible values of $s$ are $s=2,4,5$. (None of $1,3,6,7,8,9$ is a divisor of 500.) We know that $r$ is a multiple of $s$, that $r>s$ (because $s$ is the remainder when $n$ is divided by $r$ ), and that $1 \leq r \leq 15$. If $s=5$, then $r=10$ or $r=15$. If $s=4$, then $r=8$ or $r=12$. If $s=2$, then $r=4,6,8,10,12,14$. Suppose that $s=5$ and $r=10$. Since $m=d n+r$, then $500=d n+10$ and so $d n=490$. Therefore, $n$ is a divisor of 490, is a multiple of 5 (because $n=c s$ ), must be greater than $r=10$, and must be 5 more than a multiple of 10 (because the remainder when $n$ is divided by $r$ is $s$ ). Since $490=5 \times 2 \times 7^{2}$, then the divisors of 490 that are multiples of 5 are $5,10,35,70,245,490$ (these are 5 times the divisors of $2 \times 7^{2}$ ). Among these, those greater than $r=10$ having remainder 5 when divided by 10 are 35 and 245, and so the possible values of $n$ in this case are 35 and 245. For each possible pair $s$ and $r$, we determine the values of $n$ that satisfy the following conditions: - $n$ is a divsior of $500-r$, - $n$ is a multiple of $s$, - $n$ is greater than $r$, and - the remainder when $n$ is divided by $r$ is $s$. Therefore, the possible values of $n$ are $35,245,485,12,164,492,244,62,26,38,494,82,122$, of which there are 13.