What is the percentage of odd composite positive numbers divisible by 3?
In that same vein, what is the percentage of odd composite positive numbers divisible by 5?
And, for the future, what is the percentage of odd composite positive numbers divisible by n where n is some prime greater than 2?
Best Answer
"Percentage" here is not a very precise term. So we give a precise version of your question. Let $D(n)$ be the number of odd composites $\le n$ that are divisible by $3$. Let $OC(n)$ be the number of odd composites $\le n$. We want to find $$\lim_{n\to\infty} \frac{D(n)}{OC(n)}.$$ This is $$\lim_{n\to\infty} \frac{D(n)}{n}\cdot \frac{n}{OC(n)}.$$ Every sixth number after $3$ is an odd composite divisible by $3$, so $$\lim_{n\to\infty}\frac{D(n)}{n}=\frac{1}{6}.$$ By the Prime Number Theorem, $$\lim_{n\to\infty}\frac{n}{OC(n)}=2$$ (after a while, primes are sparse). Thus the required limit is $\frac{1}{3}$.