hey ! try to plug in let k = 4, t=3
we took these number becoz it 4^2 - 3^2= 7 which is an odd integer.
a) k + t + 2 plug in 4+3+2=9 which is odd
b) k^2 +2kt +t^2 plug in 16+2*4*3+9=49 which is odd
c) k^2 + t^2 plug in 16+9=25 which is odd
hence answer is none
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Ian Stewart
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k^2 - t^2 is odd. what must be odd:
I) k+t+2
II) k^2 + 2kt + t^2
III) k^2 + t^2
There are a few ways to look at this. First, when dealing with integers, notice that addition and subtraction follow the same rules where evens and odds are concerned. That is,
if a+b is even, then a-b is even
if a+b is odd, then a-b is odd
So if k^2 - t^2 is odd, so is k^2+t^2. So III) is definitely odd. From here you can see that II) must also be odd, but I think it's more illuminating if you factor: k^2 - t^2 is a difference of squares, so
k^2 - t^2 = (k+t)(k-t)
We know (k+t)(k-t) is odd, and the only way you can have an odd product of integers is if those integers are odd. Thus k+t and k-t are odd. From there, I) must be odd, and II) must be as well, since it's the same as (k+t)^2. This question is partly testing whether you recognize two of the most common 'factoring patterns': a^2 - b^2 and a^2 + 2ab + b^2.
I) k+t+2
II) k^2 + 2kt + t^2
III) k^2 + t^2
There are a few ways to look at this. First, when dealing with integers, notice that addition and subtraction follow the same rules where evens and odds are concerned. That is,
if a+b is even, then a-b is even
if a+b is odd, then a-b is odd
So if k^2 - t^2 is odd, so is k^2+t^2. So III) is definitely odd. From here you can see that II) must also be odd, but I think it's more illuminating if you factor: k^2 - t^2 is a difference of squares, so
k^2 - t^2 = (k+t)(k-t)
We know (k+t)(k-t) is odd, and the only way you can have an odd product of integers is if those integers are odd. Thus k+t and k-t are odd. From there, I) must be odd, and II) must be as well, since it's the same as (k+t)^2. This question is partly testing whether you recognize two of the most common 'factoring patterns': a^2 - b^2 and a^2 + 2ab + b^2.
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VerbalAttack
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Another alternative is to simply even further;
k^2 - t^2 is odd. That means either k^2 or t^2 is odd and the other is even;
If k^2 is odd & t^2 is even ==> k is odd & t is even
If k^2 is even & t^2 is odd ==> k is even & t is odd
We assume k is odd & t is even;
I) k + t + 2 ==> o + e + e ==> o
II) k^2 + 2kt + t^2 ==> o + e + e ==> o
III) k^2 + t^2 ==> o + e ==> o
Answer is NONE.
Cheers
k^2 - t^2 is odd. That means either k^2 or t^2 is odd and the other is even;
If k^2 is odd & t^2 is even ==> k is odd & t is even
If k^2 is even & t^2 is odd ==> k is even & t is odd
We assume k is odd & t is even;
I) k + t + 2 ==> o + e + e ==> o
II) k^2 + 2kt + t^2 ==> o + e + e ==> o
III) k^2 + t^2 ==> o + e ==> o
Answer is NONE.
Cheers
