For $N\in\mathbb N$, let $$P_l(N)=\# \{(n,m)|0\le n\le N, 0\le m\le n,\binom{n}{m}\not\equiv 0 \mod l\}.$$ Suppose that $\binom{n}{0}=1$ for $n\ge 0$ and that $\# S$ represents the number of the elements of a set $S$.

Then, here are my questions.

**Question 1** : Find $\lim_{N\to\infty}\log_N P_6(N)$.

**Question 2** : Find $\lim_{N\to\infty}\log_N P_l(N)$ for $l={p_1}^{m_1}{p_2}^{m_2}\cdots {p_s}^{m_s}\ (p_1\lt p_2\lt \cdots\lt p_s)$.

**Motivation** : I've found that $\lim_{N\to\infty}\log_N P_p(N)=\log_p \frac{p(p+1)}{2}$ for any prime number $p$. However, I'm facing difficulty for non-prime-number cases. Can anyone help?

**Remark** : This question has been asked previously on math.SE without receiving any answers.