{
  "id": "1510.06696",
  "version": "v1",
  "published": "2015-10-22T17:43:00.000Z",
  "updated": "2015-10-22T17:43:00.000Z",
  "title": "On the Greatest Common Divisor of $\\binom{qn}{q}, \\binom{qn}{2q},\\dots, \\binom{qn}{qn-q}$",
  "authors": [
    "Carl McTague"
  ],
  "comment": "2 pages",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Every binomial coefficient aficionado knows that $\\mathrm{GCD}_{0<k<n}\\binom n k$ equals $p$ if $n=p^i$ for some $i>0$ and equals 1 otherwise. It is less well known that $\\mathrm{GCD}_{0<k<n} \\binom{2n}{2k}$ equals (a power of 2 times) the product of all odd primes $p$ such that $2n=p^i+p^j$ for some $i\\le j$. This note gives a concise proof of a tidy generalization of these facts.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-22T17:43:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A10",
      "11B65"
    ],
    "keywords": [
      "greatest common divisor",
      "binomial coefficient aficionado knows",
      "tidy generalization",
      "odd primes",
      "concise proof"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 2,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}