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FoolForMath

Fool's problems of the day, [** EDIT: Complimentary problem added **] On the April fools' day I give you three cute problems on number theory/Combinatorics: \((1) \) Find the number of non-negative integral solutions of \(3x+4y=120 \) [Solved: @2bornot2b ] \( (2) \) If \(81x + 64y = n\) find the greatest possible of \( n \) such that both \( (x, y) \) are not positive. \( (3) \) Let \(n\) and \( m\) be positive integers. An \( n \times m \) rectangle is tiled with unit squares. Let \(r(n,m) \) denote the number of rectangles formed by the edge of these unit squares. Thus, for example, \(r(2, 1) = 3\). Can you find \( r(11, 12) \)?. [ Solved by @satellite73] PS: The problems are arranged in an increasing order of difficulty. (However, (2) could seem very very easy to a number theorist!) ** Complimentary for those who feels these are very hard:** If the tangent the to the curve \(\sqrt{x} + \sqrt{y} = \sqrt{a} \) at any point on cuts x-axis and y-axis at two distinct points. Can you find the sum of the intercepts ? Good luck!

  • 2 years ago
  • 2 years ago

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  1. Ishaan94
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    |dw:1333293300537:dw|

    • 2 years ago
  2. Ishaan94
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    Why did everyone leave? :(

    • 2 years ago
  3. Ishaan94
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    Now how do I get the number of positive integral solutions? I don't wanna use hit and trial method :/ We know \(y\le30\) and \(x\le40\).

    • 2 years ago
  4. FoolForMath
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    HINT: Linear Diophantine equation.

    • 2 years ago
  5. lgbasallote
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    these problems would be unsolveable and fool will greet us April Fool's Day mmhmm <nods> that would be epic

    • 2 years ago
  6. FoolForMath
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    (1) is very much solvable by almost anybody.

    • 2 years ago
  7. experimentX
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    both of \( (x, y) \) are not positive??? either x is negative or y is negative or both are negative?

    • 2 years ago
  8. experimentX
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    looks like n is greatest common divisor

    • 2 years ago
  9. Ishaan94
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    Diophantine sounds like organic chemistry. I'm googling it. 10?

    • 2 years ago
  10. FoolForMath
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    Lol, he (Diophantus) is the sometimes called the father of algebra.

    • 2 years ago
  11. Ishaan94
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    It took me time but the Linear Diophantine thing is really nice. Thanks.

    • 2 years ago
  12. experimentX
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    is the answer of 2 1??

    • 2 years ago
  13. FoolForMath
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    1 is the not the right answer.

    • 2 years ago
  14. AravindG
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    ......

    • 2 years ago
  15. experimentX
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    must be infinity then

    • 2 years ago
  16. FoolForMath
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    No it's not infinity.

    • 2 years ago
  17. 2bornot2b
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    There are 11 non negative integral solutions to the given diophantine equation

    • 2 years ago
  18. 2bornot2b
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    Am I right?

    • 2 years ago
  19. FoolForMath
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    Well done @2bornot2b!

    • 2 years ago
  20. 2bornot2b
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    Thanks!

    • 2 years ago
  21. FoolForMath
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    If other want, you may like to the post the solution too.

    • 2 years ago
  22. 2bornot2b
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    Using euclidean reduction it can be shown that the integral solutions are of the form \(x=4n\) and \(y=-3(n-10)\) where n is any integer. One can also use the formula to solve diophantine equation or any other method. Now for positive n, 4n is always positive, so we can't take into account any negative n. Again the expression (n-10) must be negative so as to have positive y, which is possible for n=0,1,2,3....,10. So we have 11 solutions

    • 2 years ago
  23. 2bornot2b
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    And yes for n-10=0, the value is true too.

    • 2 years ago
  24. Ishaan94
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    For 2nd 81*64?

    • 2 years ago
  25. FoolForMath
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    @Ishaan94: Ishaan you are very close.

    • 2 years ago
  26. Ishaan94
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    81*64 - 1? lol

    • 2 years ago
  27. FoolForMath
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    haha, nopes :)

    • 2 years ago
  28. Ishaan94
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    0?

    • 2 years ago
  29. experimentX
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    i was thinking the same .. lol

    • 2 years ago
  30. experimentX
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    but it seems 0 is nowhere close to 81*64

    • 2 years ago
  31. Ishaan94
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    Yeah lol

    • 2 years ago
  32. Mr.Math
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    Isn't it obvious that if both \(x\) and \(y\) are non-positive, then the largest possible value of \(n\) is \(0\)?

    • 2 years ago
  33. Mr.Math
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    I'm talking about the second problem.

    • 2 years ago
  34. experimentX
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    yeah ... but both \( (x, y) \) are not positive. might mean both cannot be ++, but can be +- or -+ ... just considering possibility

    • 2 years ago
  35. Ishaan94
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    It is, but at first I presumed it was both x and y can't be non-positive integral :/

    • 2 years ago
  36. Ishaan94
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    If you'll read the first question you might understand the cause of my presumption :/

    • 2 years ago
  37. experimentX
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    still, considering this http://en.wikipedia.org/wiki/Diophantine_equation#Linear_Diophantine_equations i still think 0 is the answer, since 81and 64 do not have any common divisor

    • 2 years ago
  38. Mr.Math
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    Oh he probably meant "such that x and y are not both positive", meaning that only one of them can be positive.

    • 2 years ago
  39. experimentX
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    in that case .. Mr. Math must be right.

    • 2 years ago
  40. Ishaan94
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    We need to minimize the indicated portion, which is only possible if \(n\le 0\), and 0 is the largest as the rest of the values are negative. |dw:1333297533937:dw|

    • 2 years ago
  41. Mr.Math
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    The answer of (2) is \(0\), unless something's wrong with the question.

    • 2 years ago
  42. Ishaan94
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    FoolForMath for the 3rd question, Can you tell the value of r(3,2)?

    • 2 years ago
  43. experimentX
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    @FoolForMath what do you say??

    • 2 years ago
  44. Ishaan94
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    FoolForMath WHAT SAY YOU? *This is more cinematic hehe*

    • 2 years ago
  45. Mr.Math
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    What does "is tiled with" mean? (My English is failing me)!

    • 2 years ago
  46. Ishaan94
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    Hmm made up of, maybe

    • 2 years ago
  47. Mr.Math
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    Oh I see.

    • 2 years ago
  48. Mr.Math
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    But why is \(r(2,1)=3\)?

    • 2 years ago
  49. FoolForMath
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    Ishaan, \( r(3,2)=18 \).

    • 2 years ago
  50. satellite73
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    no you are trying to find n is r(3,2)=16?

    • 2 years ago
  51. Mr.Math
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    @satellite, if both \(x\) and \(y\) are not positive then obviously the maximum value of \(n\) is \(0\). If only one of them has to be non-positive, then we can choose \(x>>> 0\ge y\) and then n would have no maximum.

    • 2 years ago
  52. experimentX
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    number of rectangles?? you mean there are lot's of rectangle???

    • 2 years ago
  53. satellite73
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    not both postive ok i see r(3,2)=18

    • 2 years ago
  54. FoolForMath
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    I think both \( x \) and \( y\) are not positive means either of them can be positive and both of them can be negative. x=1, y=-1 then, 81-64 = 17 you have to maximize this value.

    • 2 years ago
  55. satellite73
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    ?? lost me on that one

    • 2 years ago
  56. experimentX
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    x=inf, y=-1, n = inf

    • 2 years ago
  57. satellite73
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    since gcd(81,64)=1 we can solve \[81x+64y=1\] for x and y and so can solve \[81x+64y=n\] for any n

    • 2 years ago
  58. Ishaan94
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    But as long as you have positive y intercept and x intercept, you will always have positive solutions for (x,y) subject to \(x,y \in \mathbb{R}\).

    • 2 years ago
  59. FoolForMath
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    sat, if x and y are non-negative integers, then the the greatest integer that cannot be written in the form ax+by is ab − a − b. [assuming (a,b)=1 ]

    • 2 years ago
  60. experimentX
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    x=inf, y=-1, n = inf what about this ??

    • 2 years ago
  61. satellite73
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    i have an answer i like for the second to last one

    • 2 years ago
  62. satellite73
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    whew. left the house, had a beer and thought about it a little more clearly

    • 2 years ago
  63. FoolForMath
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    Congratz sat! :) You may like to post your solution for others.

    • 2 years ago
  64. satellite73
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    not unless everyone is done

    • 2 years ago
  65. FoolForMath
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    Alright:) Third one is the hardest in my opinion. So well done agian! :)

    • 2 years ago
  66. satellite73
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    merci

    • 2 years ago
  67. anonymoustwo44
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    uhm for number 2, do you mean the greatest possible value of n for x<0 and y<0?

    • 2 years ago
  68. bluepig148
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    Is the answer to the rectangle one 5148?

    • 2 years ago
  69. bluepig148
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    @satellite73 Am I right?

    • 2 years ago
  70. satellite73
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    that is what i got, yes

    • 2 years ago
  71. bluepig148
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    It's been a while, so http://ideone.com/k7o4g My thought process was explained in the comments.

    • 2 years ago
  72. Ishaan94
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    For the complimentary problem, \(a^2\)?

    • 2 years ago
  73. FoolForMath
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    @Ishaan94: \(a^2 \) is not the right answer.

    • 2 years ago
  74. lgbasallote
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    @FoolForMath do you make these things up? or you have some referneces? coz you would be sooo smart if these are original -___-

    • 2 years ago
  75. perl
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    If the tangent the to the curve x√+y√=a√ at any point on cuts x-axis and y-axis at two distinct points. Can you find the sum of the intercepts ?

    • 2 years ago
  76. .Sam.
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    hmm

    • 2 years ago
  77. .Sam.
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    answer for (2) is it n=-1?

    • 2 years ago
  78. .Sam.
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    or it could be zero

    • 2 years ago
  79. FoolForMath
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    #2 answer : 5039 Complimentary problem: \( a \)

    • 2 years ago
  80. Ishaan94
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    Oh yeah, a

    • 2 years ago
  81. experimentX
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    explanation???

    • 2 years ago
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