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If k and t are integers and k^2 – t^2 is an odd integer

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If k and t are integers and k^2 – t^2 is an odd integer [#permalink] New post 02 Oct 2007, 12:27
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81% (02:01) correct 19% (01:09) wrong based on 180 sessions
If k and t are integers and k^2 – t^2 is an odd integer, which of the following must be an even integer?

I. k + t + 2
II. k^2 + 2kt + t^2
III. k^2 + t^2

A. None
B. I only
C. II only
D. III only
E. I, II, and III
[Reveal] Spoiler: OA
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Re: GMAT Prep 2: PS Q25 [#permalink] New post 02 Oct 2007, 12:33
avenger wrote:
Q25)
If k and t are integers and k^2 – t^2 is an odd integer, which of the following must be an even integer?

I. k + t + 2
II. k^2 + 2kt + t^2
III. k^2 + t^2

(a) None
(b) I only
(c) II only
(d) III only
(e) I, II, and III


For k^2 – t^2 to be odd , I guess either k or t must be odd .
So lets assume k= even t = odd

I. k + t + 2
even + odd + 2 = even + odd = odd

II. k^2 + 2kt + t^2
even + even + odd = even + odd = odd

III. k^2 + t^2
even + odd = odd .

So I guess NONE
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 [#permalink] New post 02 Oct 2007, 12:41
none..i.e E

take any example.. 3 & 4 and plug it in each option.
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 [#permalink] New post 02 Oct 2007, 14:45
OA is A i.e. (None).

Thanks
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Re: Q25) If k and t are integers and k^2 t^2 is an odd [#permalink] New post 07 May 2012, 07:42
avenger wrote:
Q25)
If k and t are integers and k^2 – t^2 is an odd integer, which of the following must be an even integer?

I. k + t + 2
II. k^2 + 2kt + t^2
III. k^2 + t^2

(a) None
(b) I only
(c) II only
(d) III only
(e) I, II, and III


I have a doubt regarding option(III) of the stem,below is my explanation:
Given : K^2-t^2--> odd
it means (k+t)(k-t) both are odd.
take option 3 we have to chek whether k^2+t^2 is odd or even.
k^2+t^2=(k+t)^2-(k-t)^2
= k^2+t^2+2kt-k^2-t^2+2kt
=4kt
Here as 4 is an even number, and any odd number multiplied by an even results in an even number.

Please let me know whether this is correct as i had interpreted or not. and provide a suitable explanation.
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Re: Q25) If k and t are integers and k^2 t^2 is an odd [#permalink] New post 07 May 2012, 08:23
Expert's post
piyushksharma wrote:
avenger wrote:
Q25)
If k and t are integers and k^2 – t^2 is an odd integer, which of the following must be an even integer?

I. k + t + 2
II. k^2 + 2kt + t^2
III. k^2 + t^2

(a) None
(b) I only
(c) II only
(d) III only
(e) I, II, and III


I have a doubt regarding option(III) of the stem,below is my explanation:
Given : K^2-t^2--> odd
it means (k+t)(k-t) both are odd.
take option 3 we have to chek whether k^2+t^2 is odd or even.
k^2+t^2=(k+t)^2-(k-t)^2
= k^2+t^2+2kt-k^2-t^2+2kt
=4kt

Here as 4 is an even number, and any odd number multiplied by an even results in an even number.

Please let me know whether this is correct as i had interpreted or not. and provide a suitable explanation.


The red part is not correct. k^2+t^2 does not equal to 4kt.

If k and t are integers and k^2 – t^2 is an odd integer, which of the following must be an even integer?

I. k + t + 2
II. k^2 + 2kt + t^2
III. k^2 + t^2

A. None
B. I only
C. II only
D. III only
E. I, II, and III

k^2–t^2 is an odd integer means that either k is even and t is odd or k is odd and t is even.

Check all options:
I. k + t + 2 --> even+odd+even=odd or odd+even+even=odd. Discard;
II. k^2 + 2kt + t^2 --> even+even+odd=odd or odd+even+even=odd. Discard;
III. k^2 + t^2 --> even+odd=odd or odd+even=odd. Discard.

Answer: A.
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Re: Q25) If k and t are integers and k^2 t^2 is an odd [#permalink] New post 07 May 2012, 09:54
Bunuel wrote:
piyushksharma wrote:
avenger wrote:
Q25)
If k and t are integers and k^2 – t^2 is an odd integer, which of the following must be an even integer?

I. k + t + 2
II. k^2 + 2kt + t^2
III. k^2 + t^2

(a) None
(b) I only
(c) II only
(d) III only
(e) I, II, and III


I have a doubt regarding option(III) of the stem,below is my explanation:
Given : K^2-t^2--> odd
it means (k+t)(k-t) both are odd.
take option 3 we have to chek whether k^2+t^2 is odd or even.
k^2+t^2=(k+t)^2-(k-t)^2
= k^2+t^2+2kt-k^2-t^2+2kt
=4kt

Here as 4 is an even number, and any odd number multiplied by an even results in an even number.

Please let me know whether this is correct as i had interpreted or not. and provide a suitable explanation.


The red part is not correct. k^2+t^2 does not equal to 4kt.

If k and t are integers and k^2 – t^2 is an odd integer, which of the following must be an even integer?

I. k + t + 2
II. k^2 + 2kt + t^2
III. k^2 + t^2

A. None
B. I only
C. II only
D. III only
E. I, II, and III

k^2–t^2 is an odd integer means that either k is even and t is odd or k is odd and t is even.

Check all options:
I. k + t + 2 --> even+odd+even=odd or odd+even+even=odd. Discard;
II. k^2 + 2kt + t^2 --> even+even+odd=odd or odd+even+even=odd. Discard;
III. k^2 + t^2 --> even+odd=odd or odd+even=odd. Discard.

Answer: A.

Thanks, i misinterpreted the option (III),rather i solved it for k^2 - t^2.
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Re: If k and t are integers and k^2 – t^2 is an odd integer [#permalink] New post 27 Mar 2013, 03:51
i have chosen to solve using fact that since k^2 - t^2 is odd, both K+T and K-T should be odd.
Making this choice, i get that both options 1 and 2 are odd
1) k+t+2 means odd number + 2 = odd number
2) (k+t)^2 means (odd)^2 = odd number
3) k^2 + t^2 = ((k+t)^2 + (k-t)^2)/2 => (odd + odd)/2 = even

so, result is (1) and (2) are odd while (3) is even, since this combination is not part of any answer, chose NONE.

Is this approach correct?
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Re: If k and t are integers and k^2 – t^2 is an odd integer [#permalink] New post 27 Mar 2013, 05:36
Expert's post
mamathak wrote:
i have chosen to solve using fact that since k^2 - t^2 is odd, both K+T and K-T should be odd.
Making this choice, i get that both options 1 and 2 are odd
1) k+t+2 means odd number + 2 = odd number
2) (k+t)^2 means (odd)^2 = odd number
3) k^2 + t^2 = ((k+t)^2 + (k-t)^2)/2 => (odd + odd)/2 = even

so, result is (1) and (2) are odd while (3) is even, since this combination is not part of any answer, chose NONE.

Is this approach correct?


No, that's NOT correct.

k^2–t^2 is an odd integer means that either k is even and t is odd or k is odd and t is even.

Check here: if-k-and-t-are-integers-and-k-2-t-2-is-an-odd-integer-53248.html#p1082758

Hope it helps.
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Re: If k and t are integers and k^2 – t^2 is an odd integer [#permalink] New post 27 Mar 2013, 05:50
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If k and t are integers and k^2 – t^2 is an odd integer, which of the following must be an even integer?

I. k + t + 2
II. k^2 + 2kt + t^2
III. k^2 + t^2

A. None
B. I only
C. II only
D. III only
E. I, II, and III

k^2 -t^2 = odd
so,if k is even,t will be odd.

I. k+t+2 = even+odd+2 = odd
II. k^2 + 2kt+t^2 = even + even + odd = odd
III. k^2 + t^2 = even + odd = odd

Hence answer will be option A.
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Re: If k and t are integers and k^2 – t^2 is an odd integer [#permalink] New post 27 Mar 2013, 07:28
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This problem mobilizes addition, substraction and multiplication rules of odd and even integers so knowing your rules will be the key to helping you solve this correctly. :)

First of all, k^2 - t^2 can be rewritten as (k-t)*(k+t)

If you're unsure about the notation, just develop (k-t)*(k+t).

Now, assuming that k^2 - t^2 = (k-t)*(k+t) is odd, then according to the following rule :

odd * odd = odd (1)

We'll get k-t is odd and k+t is odd, which is extremely helpful since, if you notice the answer choices, all of them revolve around k+t. So let's go through them one by one :

I. k+t+2.

Using parenthesis to isolate k+t, we get (k+t)+2 which is a sum involving an odd number and an even number. So, according to the following rule :

odd + even = odd (2)

Which means that k+t+2 is odd. So answer I is not possible. (Since we're looking for an even result)

II. k^2 + 2kt + t^2

Now this answer choice may seem intimidating, but it actually isn't. Since k^2 + 2kt + t^2 is equal to (k+t)^2. And since k+t is odd, then its square will be odd as well (rule 1). So answer II is also not possible.

III.k^2 + t^2

Once again, this answer choice may seem intimidating since you have no data on k nor t. But, looking at answer choice II., k^2+t^2 is actually equal to (k+t)^2 - 2kt.
This is a difference between an odd number (k+t)^2 and an even number 2kt, so according to rule 2, the result will be odd. So answer III. is also not possible.

As such, the only correct answer to this question is answer A.

Hope that helped. :)
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Re: Q25) If k and t are integers and k^2 t^2 is an odd [#permalink] New post 04 Sep 2013, 03:18
Bunuel wrote:
piyushksharma wrote:
avenger wrote:
Q25)
If k and t are integers and k^2 – t^2 is an odd integer, which of the following must be an even integer?

I. k + t + 2
II. k^2 + 2kt + t^2
III. k^2 + t^2

(a) None
(b) I only
(c) II only
(d) III only
(e) I, II, and III


I have a doubt regarding option(III) of the stem,below is my explanation:
Given : K^2-t^2--> odd
it means (k+t)(k-t) both are odd.
take option 3 we have to chek whether k^2+t^2 is odd or even.
k^2+t^2=(k+t)^2-(k-t)^2
= k^2+t^2+2kt-k^2-t^2+2kt
=4kt

Here as 4 is an even number, and any odd number multiplied by an even results in an even number.

Please let me know whether this is correct as i had interpreted or not. and provide a suitable explanation.


The red part is not correct. k^2+t^2 does not equal to 4kt.

If k and t are integers and k^2 – t^2 is an odd integer, which of the following must be an even integer?

I. k + t + 2
II. k^2 + 2kt + t^2
III. k^2 + t^2

A. None
B. I only
C. II only
D. III only
E. I, II, and III

k^2–t^2 is an odd integer means that either k is even and t is odd or k is odd and t is even.

Check all options:
I. k + t + 2 --> even+odd+even=odd or odd+even+even=odd. Discard;
II. k^2 + 2kt + t^2 --> even+even+odd=odd or odd+even+even=odd. Discard;
III. k^2 + t^2 --> even+odd=odd or odd+even=odd. Discard.

Answer: A.


Hi Bunuel,
You've mentioned that if k^2–t^2 is an odd integer means that either k is even and t is odd or k is odd and t is even.

Isn't a 3rd case also possible where K is odd and T is 0?
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Re: Q25) If k and t are integers and k^2 t^2 is an odd [#permalink] New post 04 Sep 2013, 03:26
Expert's post
emailmkarthik wrote:
Bunuel wrote:
piyushksharma wrote:
If k and t are integers and k^2 – t^2 is an odd integer, which of the following must be an even integer?

I. k + t + 2
II. k^2 + 2kt + t^2
III. k^2 + t^2

(a) None
(b) I only
(c) II only
(d) III only
(e) I, II, and III

I have a doubt regarding option(III) of the stem,below is my explanation:
Given : K^2-t^2--> odd
it means (k+t)(k-t) both are odd.
take option 3 we have to chek whether k^2+t^2 is odd or even.
k^2+t^2=(k+t)^2-(k-t)^2
= k^2+t^2+2kt-k^2-t^2+2kt
=4kt

Here as 4 is an even number, and any odd number multiplied by an even results in an even number.

Please let me know whether this is correct as i had interpreted or not. and provide a suitable explanation.


The red part is not correct. k^2+t^2 does not equal to 4kt.

If k and t are integers and k^2 – t^2 is an odd integer, which of the following must be an even integer?

I. k + t + 2
II. k^2 + 2kt + t^2
III. k^2 + t^2

A. None
B. I only
C. II only
D. III only
E. I, II, and III

k^2–t^2 is an odd integer means that either k is even and t is odd or k is odd and t is even.

Check all options:
I. k + t + 2 --> even+odd+even=odd or odd+even+even=odd. Discard;
II. k^2 + 2kt + t^2 --> even+even+odd=odd or odd+even+even=odd. Discard;
III. k^2 + t^2 --> even+odd=odd or odd+even=odd. Discard.

Answer: A.


Hi Bunuel,
You've mentioned that if k^2–t^2 is an odd integer means that either k is even and t is odd or k is odd and t is even.

Isn't a 3rd case also possible where K is odd and T is 0?


Well, since 0 is an even number, then this scenario falls into the case when k=odd and t=even.
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RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: If k and t are integers and k^2 – t^2 is an odd integer [#permalink] New post 07 Sep 2013, 23:50
Thank you, Bunuel.
Re: If k and t are integers and k^2 – t^2 is an odd integer   [#permalink] 07 Sep 2013, 23:50
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