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saifoo.khan Group Title

The sum of "z" consecutive integers is 90. Which of the following cannot be the value of "z". 3, 4, 5, 6, 9.

  • 2 years ago
  • 2 years ago

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  1. mathslover Group Title
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    it seems that the answer given by saurav is very correct

    • 2 years ago
  2. saifoo.khan Group Title
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    i still don't get it. lol.

    • 2 years ago
  3. myko Group Title
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    maybe first put it like this: \[n+(n+1)+(n+2)+...+(n+(z-1))=90\] and find it from here

    • 2 years ago
  4. saifoo.khan Group Title
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    How? Why? @myko

    • 2 years ago
  5. mukushla Group Title
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    guys lets work it step by step \[n+1+n+2+...+n+z=90\]\[nz+\frac{z(z+1)}{2}=90\]\[z^2+(2n+1)z=180\]\[z(z+2n+1)=180\] we can find all integers \(z\) that satisfies the last equality...but here we have options and we can plug them into equation and see what happens...

    • 2 years ago
  6. myko Group Title
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    the expretion would be equal to: n*z+ sum of first (z-1) integers = 90

    • 2 years ago
  7. myko Group Title
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    @mukushla your sum is of z+1 consecutive integers

    • 2 years ago
  8. mukushla Group Title
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    no its z n+1 n+2 ... n+z

    • 2 years ago
  9. myko Group Title
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    should be nz+z(z-1)/2=90

    • 2 years ago
  10. saifoo.khan Group Title
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    So..?

    • 2 years ago
  11. myko Group Title
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    2nz+z^2-z=180 z^2+(2n-1)z-180=0 and I think just check for possible natural solutions of this

    • 2 years ago
  12. mukushla Group Title
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    now u have \[n=\frac{90}{z}-\frac{z+1}{2}\] now plug the values of \(z\) and if the value of \(n\) becomes an integer then that \(z\) is your answer

    • 2 years ago
  13. saifoo.khan Group Title
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    @mukushla : Your's and @myko 's methods are same right?

    • 2 years ago
  14. myko Group Title
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    yes they are

    • 2 years ago
  15. mukushla Group Title
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    for example \(z=3\) gives \[n=\frac{90}{3}-\frac{3+1}{2}=30-2=28\]

    • 2 years ago
  16. mukushla Group Title
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    yes they are same

    • 2 years ago
  17. myko Group Title
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    only difference is that i start fomr n and @mukushla from n+1

    • 2 years ago
  18. saifoo.khan Group Title
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    So answer is 3?

    • 2 years ago
  19. myko Group Title
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    it asks: Which of the following cannot be the value of "z". and 3 aparently , is.

    • 2 years ago
  20. saifoo.khan Group Title
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    But it's incorrect. :S

    • 2 years ago
  21. mukushla Group Title
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    let me complete the solution \[z=3 ---------> n=\frac{90}{3}-\frac{3+1}{2}=28\] \[z=4 ---------> n=\frac{90}{4}-\frac{4+1}{2}=20\] \[z=5 ---------> n=\frac{90}{5}-\frac{5+1}{2}=15\] \[z=6 ---------> n=\frac{90}{6}-\frac{6+1}{2}=11.5\] \[z=9 ---------> n=\frac{90}{9}-\frac{9+1}{2}=5\] so the answer is only z=6

    • 2 years ago
  22. myko Group Title
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    4and 6 would be the answer

    • 2 years ago
  23. myko Group Title
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    i think

    • 2 years ago
  24. saifoo.khan Group Title
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    Myko's thinking is going the correct way!

    • 2 years ago
  25. myko Group Title
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    i got it by mine first formula using n like a start integer.

    • 2 years ago
  26. myko Group Title
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    but aparently there should be no difference in starting from n+1 or n+2 or n+whatever

    • 2 years ago
  27. myko Group Title
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    hmm

    • 2 years ago
  28. mukushla Group Title
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    for \(z=4\) \[21+22+23+24=90\]

    • 2 years ago
  29. rsadhvika Group Title
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    4 90/4 - 4+1/2 90/4 - 10/4 80/4 is an integer

    • 2 years ago
  30. saifoo.khan Group Title
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    4 is incorrect too!

    • 2 years ago
  31. mukushla Group Title
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    its impossible ... see my last reply

    • 2 years ago
  32. saifoo.khan Group Title
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    The answer is 6. :S

    • 2 years ago
  33. saifoo.khan Group Title
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    I'm going to type the solution.. just a sec.

    • 2 years ago
  34. mukushla Group Title
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    @saifoo.khan thats right answer is 6...

    • 2 years ago
  35. saifoo.khan Group Title
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    How? Why? :D

    • 2 years ago
  36. mukushla Group Title
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    man... we already solved it...

    • 2 years ago
  37. saifoo.khan Group Title
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    But you were getting 11.5 when we inserted 6.

    • 2 years ago
  38. mukushla Group Title
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    well that means there is not 6 consecutive integers such that their sum is 90

    • 2 years ago
  39. saifoo.khan Group Title
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    (behind the book) Solution: Plug in numbers! This may take some time, but we will eventually find the answer. Three numbers 20+30+31=90,eliminate A, four numbers 21+22+23+24=90,eliminate b,five numbers 16+17+18+19+20=90,elimnate c.nine numbers 6+7+8+9+10+11+12+13+14=90 elimate e.no six consecutive numbers add up to 90

    • 2 years ago
  40. saifoo.khan Group Title
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    @mukushla , got it right!

    • 2 years ago
  41. saifoo.khan Group Title
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    I totally get it. Thanks a ton! @mukushla & @myko

    • 2 years ago
  42. mukushla Group Title
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    yw :)

    • 2 years ago
  43. myko Group Title
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    yw

    • 2 years ago
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