Question Bank - Number of Integral/Positive/Non-Negative Solutions - Set 2


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    24 = 2^3 * 3
    a * b * c = 24
    => if a = 2^m * 3^n
    b = 2^p * 3^q
    c = 2^x * 3^y
    We can say that m + p + x = 3 and n + q + y = 1
    So total solutions = (3 + 3 - 1)C(3 - 1) * (1 + 3 - 1)C(3 - 1)
    = 5C2 * 3C2
    = 10 * 3
    = 30 ways


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    We are basically looking for perfect squares between 6 and 101.
    9, 16, 25, 36, 49, 64, 81, 100
    8 values.


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    (x+y)^(y-1) = 100

    100 can be represented in p^q form in 2 ways.

    case 1: 10^2
    (x,y) = (7,3)

    case 2: 100^1
    (x,y) = (98,2)

    So 2 ways.


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    2009 = 41 * 49
    √2009 = 7√41
    as √41 is irrational it has to be present in both √x and √y.
    so we can write, a√41 + b√41 = 7√41
    a + b = 7
    3 pairs will satisfy - (4, 3), (5, 2) and (6, 1)


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    a^2 – b^2 = 288
    288 = 2^5 * 3^2
    Integer solutions = 2 * Number of factors of (288/4)
    288/4 = 72 = 2^3 * 3^2
    Number of factors = 4 * 3 = 12
    Integer solutions = 2 * 12 = 24


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    110 = 2 * 5 * 11
    Number of ordered positive triplets = (1 + 3 - 1)C2 * (1 + 3 - 1)C2 * (1 + 3 - 1)C2 = 3 * 3 * 3 = 27
    Ordered integer triplets = 4 * 27 = 108 (considering the possibilities (-a, -b, c), (-a, b, -c), (a, -b, -c), (a, b, c))
    This includes cases like aab and aaa, which needs to be removed to get the unordered pairs.
    a = b = c : no cases possible
    a = b or b = c or c = a
    As all the prime factors only once, we cannot split it into cases where 2 entries are equal. So the only cases are
    (110, 1, 1) & (110, -1, -1) -> which can be arranged in 3!/2! = 3 ways each
    All the other triplets can be arranged in 3! ways
    So final unordered triplets = (108 - 6)/3! + 6/3 = 102/6 + 2 = 17 + 2 = 19 ways.



  • @zabeer

    a = 5
    b = -12

    so 17



  • This post is deleted!


  • @zabeer answer says 5c2 * 3c2=30 but shouldnt we reduce (1 * 1* 24, 2* 2*6) like in Q7



  • @vikas_saini can u plz explain 2nd and 3rd+4th step again. 2nd step:what is -1-what are we reducing in it
    3rd step: 3rd+4th step doesnt giv value in 2nd step. what are we missing in it. 3rd+4th= 2T-2


 

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