# Some divisibility properties of binomial
and *q*-binomial coefficients

### (16 pages)

**Abstract.**
We first prove that if *a* has a prime
factor not dividing *b* then there are infinitely many
positive integers *n* such that
is not divisible by *bn*+1.
This confirms a recent conjecture of
Z.-W. Sun.
Moreover, we provide some new divisibility properties of binomial
coefficients: for example, we prove that
and
are divisible
by 6n-1, and that
is divisible by 66*n*-1, for all
positive integers *n*. As we show, the latter results are in fact
consequences of divisibility and positivity results for
quotients of *q*-binomial coefficients by *q*-integers,
generalising the positivity of *q*-Catalan numbers.
We also put forward several related conjectures.

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