Prime powers in integers, least common multiples, factorials and binomial coefficients

The following theorems are usually considered so elementary by professional mathematicians that they are stated without proof.  For hobbyists like me, to whom they may not be so transparent, proofs are provided in the attached pdf.  In what follows (1) indpn represents the index of the prime p in the prime composition of n – for example ind2 24 = 3, ind3 24 = 1, ind5 24 = 0  (2) [n] is the largest integer less than or equal to n and (3) Ln is the least positive integer divisible by each of 1, 2, … , n.

Theorem \: 1: \: ind_{p} \:n \le \ln n / \ln p

\displaystyle Theorem \: 2: \: ind_{p}\: L_{n} =\left[\frac{\ln n}{\ln p} \right]

\displaystyle Theorem \: 3: \: ind_{p} \: n!=\sum_{m \geq 1} \left[\frac{n}{p^m} \right]

\displaystyle Theorem \:4: \: ind_{p} \: \binom {n}{k} \leq \left[ \frac {\ln n}{\ln p} \right] - ind_{p} \: k

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