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anonymous

  • 5 years ago

prove that the product of three consecutive positive integers is divisible by 2. (This has to be proved using Euclid's Division Algorithm). I hv solved it, but just want to reconfirm....

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  1. anonymous
    • 5 years ago
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    ? why three integer ? two is enough

  2. anonymous
    • 5 years ago
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    Well, the question has been put up like that....

  3. dumbcow
    • 5 years ago
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    the product of an odd integer and even will always be even and the product of an even integer with any other integer is always even and an even integer is divisible by 2 by definition (2n)*(2n+1)*(2n) = 2*[n*(2n+1)(2n)] id have to look up euclids algorithm though

  4. anonymous
    • 5 years ago
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    yeah, u hv given a trimmed version of what i did basically u hv used Euclid's div lemma only

  5. dumbcow
    • 5 years ago
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    oh cool, i didnt know if i remembered it or not you are probably correct then

  6. anonymous
    • 5 years ago
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    Euclid's division lemma says "Given two positive integers a and b, there exist unique integers q and r satisfying a = bq + r where \[0\le r < b\]

  7. anonymous
    • 5 years ago
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    Therefore, taking b as 2 we get a = 2q + r and \[0\le r < 2\]

  8. anonymous
    • 5 years ago
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    So, if r = 0, then a = 2q + 0 = 2q If r = 1, then a = 2q + 1

  9. anonymous
    • 5 years ago
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    Hence all positive integers are of the form 2a and 2q+1

  10. dumbcow
    • 5 years ago
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    right and if a number can be represented in the 2q form it can be said it is divisible by 2

  11. anonymous
    • 5 years ago
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    Yes, so for my question I hv two possibilities 1) 2q, 2q+1 and 2q 2) 2q+1, 2q, 2q+1

  12. anonymous
    • 5 years ago
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    In first case product is 8q^3 + 4q^2 = 2(4q^3 + 2q^2) and since it is a multiple of 2, it is divisible by 2

  13. anonymous
    • 5 years ago
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    In second case the product is 2(4q^3 + 4q^2 + q) which is again a multiple of 2 and hence divisible by 2

  14. dumbcow
    • 5 years ago
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    correct

  15. anonymous
    • 5 years ago
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    Thanks for confirming..U get a medal !!

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