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UnkleRhaukus

  • 2 years ago

The sum of any five consecutive integers is divisible by 5 . Proof:

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  1. UnkleRhaukus
    • 2 years ago
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    Proof: \[\text{If \(x\) is the smallest of the integers,}\]\[\text{the sum of 5 consecutive integers can be written as }\]\[x+(x+1)+(x+2)+(x+3)+(x+4)\]\[\qquad\qquad=5x+(1+2+3+4)\]\[\qquad\qquad=5x+10\]\[\qquad\qquad=5(x+2)\]\[\text{If this is divisible by 5, then }\]\[5|5(x+2)\]\[\text{ is a true statement.}\]\[\text{This is true when \((x+2)\) is an integer which is always }\]\[\text{ true because integers are closed under addition.}\] This proves the sum of any five consecutive integers is divisible by 5. \[\square\]

  2. UnkleRhaukus
    • 2 years ago
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    is this good?

  3. CliffSedge
    • 2 years ago
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    Works for me, but I'm pretty lenient when it comes to rigour.

  4. estudier
    • 2 years ago
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    a + (a+1) + (a+2) + (a+3) + (a+4) = 5a + 10 QED

  5. UnkleRhaukus
    • 2 years ago
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    |dw:1351169356249:dw|

  6. UnkleRhaukus
    • 2 years ago
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    you dont think i need so many words?

  7. estudier
    • 2 years ago
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    That's right...and x+2 is neither here nor there....

  8. estudier
    • 2 years ago
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    5x +10 is already divisible by 5

  9. CliffSedge
    • 2 years ago
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    Yeah, I do like the extra step of showing that 5x+10 = 5(x+2), then it's obvious that the number is a multiple of five, and hence is divisible by 5.

  10. estudier
    • 2 years ago
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    The extra step is superfluous (really really superfluous)

  11. jhonyy9
    • 2 years ago
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    n1=1 n2=2 n3=3 n4=4 n5=5 n1 +n2 +n3 +n4 +n5 =n15 /5 is true n=k suppose is true k1 +k2 +k3 +k4 +k5 =k15 /5 than for k=k+1 (k1 +1) +(k2 +1) +(k3 +1) +(k4 +1) +(k5 +1) = = (k1 +k2 +k3 +k4 +k5) +5= =5+5+5+5= =20/5 Q:E:D:

  12. jhonyy9
    • 2 years ago
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    oppinions ???

  13. estudier
    • 2 years ago
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    (a-2) + (a-1) + a + (a+1) + (a+2) = 5a QED

  14. UnkleRhaukus
    • 2 years ago
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    i like your induction @jhonyy9, but it is long

  15. jhonyy9
    • 2 years ago
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    so yes thank you but i think that is very very more ,,mathematically" thematically yes???

  16. UnkleRhaukus
    • 2 years ago
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    my method is a flow of implication of truth

  17. jhonyy9
    • 2 years ago
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    yes but here on openstudy have told me that a proof need being more mathematically,understandably and not for the last step more understandably for sombody inclusive who not know mathematics just a ,,little" this was for my,, Goldbach's and Collatz"s conjecture proofs"

  18. jhonyy9
    • 2 years ago
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    sorry on the second understandably i wann writing acceptably sorry

  19. estudier
    • 2 years ago
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    @UnkleRhaukus This is sufficient:- Let x, x+-1, x+-2 represent 5 consecutive integers Their sum is 5x for all x in Z (if x = 0 then x/5 is zero) Hence the sum of 5 consecutive integers is always divisible by 5

  20. UnkleRhaukus
    • 2 years ago
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    That is real good @estudier !

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