Proving a sufficient condition, can someone check my work

  • Thread starter mr_coffee
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  • #1
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ello ello!

I think i did this right but not sure! The directions are: Determine whether the statement is true or false. Prove the statement directly from the definitions or give a counter exmaple if it is false.

A sufficient condition for an integer to be divisble by 8 is hat it be divisble by 16.

[tex]\forall[/tex] integers n, if n is divisble by 8, then n is divisble by 16. This is a true statement.
Proof: Suppose n is an integer divisble by 8. BY definition of divisbility, n = 8k for some integer k. But, 8k = 4*2k, and 2k is an integer becuase k is. Hence n = 4*(some integer) and so n is divisble by 16.

Thanks!
 

Answers and Replies

  • #2
radou
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mr_coffee said:
...[tex]\forall[/tex] integers n, if n is divisble by 8, then n is divisble by 16. This is a true statement.
Proof: Suppose n is an integer divisble by 8. BY definition of divisbility, n = 8k for some integer k. But, 8k = 4*2k, and 2k is an integer becuase k is. Hence n = 4*(some integer) and so n is divisble by 16.

Thanks!
What exactly do you mean by 'divisible'? Is it that the quotient must be an integer?
 
  • #3
StatusX
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mr_coffee said:
Hence n = 4*(some integer) and so n is divisble by 16.
Where did this come from? You've shown n is divisible by 4, not 16.
 
  • #4
arildno
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16n=8(2n)
Maybe that could be used for something.
 
  • #5
shmoe
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mr_coffee said:
[tex]\forall[/tex] integers n, if n is divisble by 8, then n is divisble by 16. This is a true statement.
Take n=8. n is divisble by 8, but n is not divisible by 16.

You tried to prove:

"If n is divisible by 8 then it is divisible by 16".

but in fact you showed:

"if n is divisible by 8 then it is divisible by 4"


But they actually claimed:

"A sufficient condition for an integer to be divisble by 8 is hat it be divisble by 16."

In otherwords, "if n is divisible by 16 then it is divisible by 8."
 
  • #6
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Thanks guys!

OKay i rewrote it using, ""if n is divisible by 16 then it is divisible by 8.""

[tex]\forall[/tex] integers n, if n is divisble by 16, then n is divisble by 8. This is a true statement.

Proof: Suppose n is an integer divisble by 16. By definition of divsibilty, n = 16k for some integer k. But, 16k = (8)(2k), and 2k is an integer because k is. Hence n = 8(some integer) and so n is divisble by 16.

I think i had it switched around as shmoe pointed out
 

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