# necessary and sufficient conditions.

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#1
I;d appreciate if someone could check my thinking is correct here.

For integers n: 4 divides n is a ......... condition for 4 divides n^2

P = 4 divides n
Q = 4 divides n^2

P is false, counter - example: 4/5
Q is also false = 4/(3^2)

Therefore P implies Q and Q implies P is true and so the answer is "necessary and sufficient"
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14 years ago
#2
No, you've got the idea completely wrong.

If P implies Q, then P is a sufficient condition for Q (and Q is necessary for P). If Q implies P, then P is a necessary condition for Q (and Q is sufficient for P). Can you see why?

Which way round is it here?
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14 years ago
#3
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#4
(Original post by generalebriety)
No, you've got the idea completely wrong.

If P implies Q, then P is a sufficient condition for Q (and Q is necessary for P). If Q implies P, then P is a necessary condition for Q (and Q is sufficient for P). Can you see why?

Which way round is it here?

okay, so i should be thinking:

P implies Q is false because P is sufficient for Q but Q is not necessary for P
in the same way, Q implies P is false because Q is sufficient for P but P is not necessary for Q?
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14 years ago
#5
(Original post by LAMP)
okay, so i should be thinking:

P implies Q is false because P is sufficient for Q but Q is not necessary for P
"P is sufficient for Q" and "Q is necessary for P" are the same statement. They mean the same thing. You can't have one true without the other being true, and vice versa.

Think about it. Does the statement "4 divides n" imply the statement "4 divides n^2"? Does the statement "4 divides n^2" imply the statement "4 divides n"? If your answer to either of these is yes, prove it; if your answer to either is no, give a counterexample.
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14 years ago
#6
(Original post by LAMP)
okay, so i should be thinking:

P implies Q is false because P is sufficient for Q but Q is not necessary for P
in the same way, Q implies P is false because Q is sufficient for P but P is not necessary for Q?
No. You should be deciding from the specific question about divisibility whether P implies Q, and whether Q implies P.

In other words, suppose we know 4 divides n. Does it follow that 4 must divide n^2? If so, P implies Q. If not, it doesn't.

Conversely, suppose we know 4 divides n^2. Does it follow that 4 must divide n? If so, Q implies P. If not, it doesn't.
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#7
(Original post by DFranklin)
No. You should be deciding from the specific question about divisibility whether P implies Q, and whether Q implies P.

In other words, suppose we know 4 divides n. Does it follow that 4 must divide n^2? If so, P implies Q. If not, it doesn't.

Conversely, suppose we know 4 divides n^2. Does it follow that 4 must divide n? If so, Q implies P. If not, it doesn't.
So, in our case, we know 4 doesn't always divide n, to give an integer answer so it follows that 4 doesn't divide n^2 because from integers at n=3, they both give fractions. So P implies Q. Also, we know 4 doesn;t always divide n^2 to give an integer answer for n=3, then 4 divide by n will not always give an integer answer either , so Q implies P.
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14 years ago
#8
if 4|n (say n/4= k and k is an integer)

then

n^2/4 = kn another integer. i.e. If 4|n then 4|n^2

However 4|n^2 does not imply 4|n since 4|2^2 but 4 does not divide 2.

I hope I haven't said too much.
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#9
(Original post by MAR)
if 4|n (say n/4= k and k is an integer)

then

n^2/4 = kn another integer. i.e. If 4|n then 4|n^2

However 4|n^2 does not imply 4|n since 4|2^2 but 4 does not divide 2.

I hope I haven't said too much.
So what you;ve said is that if n/4 gives an integer answer then n^2/4 will always give an integer answer but not the other way around?
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14 years ago
#10
(Original post by LAMP)
So what you;ve said is that if n/4 gives an integer answer then n^2/4 will always give an integer answer but not the other way around?
That's what "4 divides n" means. If 4 divides n, then n/4 is an integer. Does n/4 being an integer imply that n^2/4 is an integer? Does n^2/4 being an integer imply n/4 is an integer?
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9 months ago
#11
We have:

P = 4 divides n
Q = 4 divides n^2

and we want to find a way to express:
- if 4 divides n then 4 divides n^2

we know that p => Q as whenever 4 divides n, 4 will also divide n^2.
proving this goes as follows:
if 4 divides n then n = 4k
therefore n^2 =16k^2
therefore n^2 = 4(4k^2)
so, n^2 has a factor of 4

as P implies Q the statement is sufficient.
-----------------------------------------------------------
to check necessity lets assume that Q implies P
so, if 4 is a factor of n^2 then 4 is a factor of n
lets take 2 for example as this provides a counter example
2^2 is a factor of 4 however, 4 is not a factor of 2,
-----------------------------------------------------------
therefore the statement is sufficient but not necessary
Last edited by Blue Bumble Bee; 9 months ago
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9 months ago
#12
(Original post by Blue Bumble Bee)
We have:P = 4 divides nQ = 4 divides n^2and we want to find a way to express: - if 4 divides n then 4 divides n^2we know that p => Q as whenever 4 divides n, 4 will also divide n^2.proving this goes as follows:if 4 divides n then n = 4ktherefore n^2 =16k^2therefore n^2 = 4(4k^2)so, n^2 has a factor of 4as P implies Q the statement is sufficient.-----------------------------------------------------------to check necessity lets assume that Q implies Pso, if 4 is a factor of n^2 then 4 is a factor of nlets take 2 for example as this provides a counter example2^2 is a factor of 4 however, 4 is not a factor of 2, -----------------------------------------------------------therefore the statement is sufficient but not necessary
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