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henpen

  • 3 years ago

How many integer n exist so that \[ \frac{n}{100-n} \] is also an integer?

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  1. henpen
    • 3 years ago
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    What is the general strategy to be adopted for 'How many integer n exist so that f(n) is also an integer?', other than bursts of insight?

  2. RadEn
    • 3 years ago
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    i think, trial error is the best way for this case

  3. sirm3d
    • 3 years ago
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    suppose that for some integer k, \[k=\frac{ n }{ 100-n }\] rearranging the equation we get \[n=\frac{ 100k }{ k+1 }\] and since k and k+1 are relatively prime, we conclude that \[(k+1)|100\]

  4. henpen
    • 3 years ago
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    What does the notation (k+1)|100 mean?

  5. sirm3d
    • 3 years ago
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    (k+1) divides 100.

  6. sirm3d
    • 3 years ago
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    when k = 0, n = 0. when k = 1, n = 50 when k = 3, n = 75

  7. sirm3d
    • 3 years ago
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    there are more values for k, but the solution set is finite.

  8. henpen
    • 3 years ago
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    OK, thanks or that

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