# Proving a sufficient condition, can someone check my work

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.

$$\forall$$ 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!

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mr_coffee said:
...$$\forall$$ 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?

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.

arildno
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Gold Member
Dearly Missed
16n=8(2n)
Maybe that could be used for something.

shmoe
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mr_coffee said:
$$\forall$$ 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."

Thanks guys!

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

$$\forall$$ 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