{
  "index": "2000-B-2",
  "type": "NT",
  "tag": [
    "NT",
    "COMB"
  ],
  "difficulty": "",
  "question": "Prove that the expression\n\\[ \\frac{gcd(m,n)}{n}\\binom{n}{m} \\]\nis an integer for all pairs of integers $n\\geq m\\geq 1$.",
  "solution": "Since $\\gcd(m,n)$ is an integer linear combination of $m$ and $n$,\nit follows that\n\\[\\frac{gcd(m,n)}{n}\\binom{n}{m}\\]\nis an integer linear combination of the integers\n\\[\\frac{m}{n}\\binom{n}{m}=\\binom{n-1}{m-1} \\mbox{ and }\n\\frac{n}{n}\\binom{n}{m}=\\binom{n}{m} \\]\nand hence is itself an integer.",
  "vars": [
    "m",
    "n"
  ],
  "params": [],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "m": "intsmall",
        "n": "intlarge"
      },
      "question": "Prove that the expression\n\\[ \\frac{gcd(intsmall,intlarge)}{intlarge}\\binom{intlarge}{intsmall} \\]\nis an integer for all pairs of integers $intlarge\\geq intsmall\\geq 1$.",
      "solution": "Since $\\gcd(intsmall,intlarge)$ is an integer linear combination of $intsmall$ and $intlarge$,\nit follows that\n\\[\\frac{gcd(intsmall,intlarge)}{intlarge}\\binom{intlarge}{intsmall}\\]\nis an integer linear combination of the integers\n\\[\\frac{intsmall}{intlarge}\\binom{intlarge}{intsmall}=\\binom{intlarge-1}{intsmall-1} \\mbox{ and }\n\\frac{intlarge}{intlarge}\\binom{intlarge}{intsmall}=\\binom{intlarge}{intsmall} \\]\nand hence is itself an integer."
    },
    "descriptive_long_confusing": {
      "map": {
        "m": "sapphire",
        "n": "pineapple"
      },
      "question": "Prove that the expression\n\\[ \\frac{gcd(sapphire,pineapple)}{pineapple}\\binom{pineapple}{sapphire} \\]\nis an integer for all pairs of integers $pineapple\\geq sapphire\\geq 1$.",
      "solution": "Since $\\gcd(sapphire,pineapple)$ is an integer linear combination of $sapphire$ and $pineapple$, it follows that\n\\[\\frac{gcd(sapphire,pineapple)}{pineapple}\\binom{pineapple}{sapphire}\\]\nis an integer linear combination of the integers\n\\[\\frac{sapphire}{pineapple}\\binom{pineapple}{sapphire}=\\binom{pineapple-1}{sapphire-1} \\mbox{ and } \\frac{pineapple}{pineapple}\\binom{pineapple}{sapphire}=\\binom{pineapple}{sapphire} \\]\nand hence is itself an integer."
    },
    "descriptive_long_misleading": {
      "map": {
        "m": "knownvalue",
        "n": "fixednumber"
      },
      "question": "Prove that the expression\n\\[ \\frac{gcd(knownvalue,fixednumber)}{fixednumber}\\binom{fixednumber}{knownvalue} \\]\nis an integer for all pairs of integers $fixednumber\\geq knownvalue\\geq 1$.",
      "solution": "Since $\\gcd(knownvalue,fixednumber)$ is an integer linear combination of $knownvalue$ and $fixednumber$, it follows that\n\\[\\frac{gcd(knownvalue,fixednumber)}{fixednumber}\\binom{fixednumber}{knownvalue}\\]\nis an integer linear combination of the integers\n\\[\\frac{knownvalue}{fixednumber}\\binom{fixednumber}{knownvalue}=\\binom{fixednumber-1}{knownvalue-1} \\mbox{ and } \\frac{fixednumber}{fixednumber}\\binom{fixednumber}{knownvalue}=\\binom{fixednumber}{knownvalue} \\]\nand hence is itself an integer."
    },
    "garbled_string": {
      "map": {
        "m": "qzxwvtnp",
        "n": "hjgrksla"
      },
      "question": "Prove that the expression\n\\[ \\frac{gcd(qzxwvtnp,hjgrksla)}{hjgrksla}\\binom{hjgrksla}{qzxwvtnp} \\]\nis an integer for all pairs of integers $hjgrksla\\geq qzxwvtnp\\geq 1$.",
      "solution": "Since $\\gcd(qzxwvtnp,hjgrksla)$ is an integer linear combination of $qzxwvtnp$ and $hjgrksla$,\nit follows that\n\\[\\frac{gcd(qzxwvtnp,hjgrksla)}{hjgrksla}\\binom{hjgrksla}{qzxwvtnp}\\]\nis an integer linear combination of the integers\n\\[\\frac{qzxwvtnp}{hjgrksla}\\binom{hjgrksla}{qzxwvtnp}=\\binom{hjgrksla-1}{qzxwvtnp-1} \\mbox{ and }\n\\frac{hjgrksla}{hjgrksla}\\binom{hjgrksla}{qzxwvtnp}=\\binom{hjgrksla}{qzxwvtnp} \\]\nand hence is itself an integer."
    },
    "kernel_variant": {
      "question": "Let b and a be integers satisfying b ≥ a ≥ 0 and b > 0.  Prove that\n\n           gcd(a,b)\n   ----------------------- · C(b , a)\n                 b\n\nis always an integer (here C(b , a)=\\binom{b}{a} denotes the usual binomial coefficient).",
      "solution": "We have to show that \n         gcd(a,b)\n  ------------------- · \\binom{b}{a}\n             b\nis an integer for every pair of integers b ≥ a ≥ 0 with b > 0.\n\nStep 1 – The easy case a = 0.\nIf a = 0 then\n  (gcd(0,b)/b)·\\binom{b}{0} = b/b · 1 = 1, \nwhich is an integer.  Hence it suffices to treat a ≥ 1 below.\n\nStep 2 – Write gcd(a,b) as an integer combination of a and b.\nBy Bézout’s identity there exist integers α and β such that\n      gcd(a,b) = α·a + β·b.\n\nStep 3 – Multiply by the binomial coefficient and divide by b.\n          gcd(a,b)          αa        βb\n  --------------------- = ----- · \\binom{b}{a} + ---- · \\binom{b}{a}\n            b               b          b\n                        = α · (a/b)\\binom{b}{a} + β · \\binom{b}{a}.\n\nStep 4 – Recognise the first factor as another binomial coefficient.\nFor a ≥ 1 the identity\n        a        b!\n (a/b)·-- = -------------- = \\binom{b-1}{a-1}\n        b   a!(b−a)!          \nshows that\n      (a/b)·\\binom{b}{a} = \\binom{b-1}{a-1} ∈ ℤ.\n\nStep 5 – Conclude integrality.\nBoth summands\n   α·\\binom{b-1}{a-1}  and  β·\\binom{b}{a}\nare integers, so their sum—and hence the original expression—is an integer.\n\nTherefore the quantity (gcd(a,b)/b)·\\binom{b}{a} belongs to ℤ for every b ≥ a ≥ 0 with b > 0. □",
      "_meta": {
        "core_steps": [
          "Apply Bézout: gcd(m,n)=αm+βn for some integers α,β",
          "Rewrite the target fraction as α·(m/n)·C(n,m)+β·C(n,m)",
          "Use the identity (m/n)·C(n,m)=C(n-1,m-1) (an integer)",
          "Both summands are integers, so their linear combination is integer"
        ],
        "mutable_slots": {
          "slot1": {
            "description": "Lower bound on m (allows m=0 without affecting proof)",
            "original": "n ≥ m ≥ 1"
          },
          "slot2": {
            "description": "Names of the two integer parameters",
            "original": "m, n"
          }
        }
      }
    }
  },
  "checked": true,
  "problem_type": "proof",
  "iteratively_fixed": true
}