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If x and y are odd positive integers, and x and y both have an odd

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If x and y are odd positive integers, and x and y both have an odd  [#permalink]

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New post 15 Sep 2019, 06:51
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If x and y are positive odd integers, and both numbers have an odd number of positive divisors, which of the following could be the value of x-y?

A) 4818
B) 5174
C) 5320
D) 5482
E) 5566

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Re: If x and y are odd positive integers, and x and y both have an odd  [#permalink]

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New post 16 Sep 2019, 08:01
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GMATPrepNow wrote:
If x and y are positive odd integers, and both numbers have an odd number of positive divisors, which of the following could be the value of x-y?

A) 4818
B) 5174
C) 5320
D) 5482
E) 5566


Key concept: Most integers have an EVEN number of positive divisors.
For example, 12 has 6 divisors, 35 has 4 divisors, and 13 has 2 divisors.
However, SQUARES of integers always have an ODD number of divisors.
For example, 25 has 3 divisors, 36 has 9 divisors, and 81 has 5 divisors.


So, if and x and y both have an odd number of positive divisors, then x and y squares of integers.
So, let’s let x = j² and let y = k² for integers j and k

NOTE: If x is odd, then j must be odd, and if y is odd, then k must be odd.

Our goal is to find a possible value of x – y
We can write: x – y = j² - k²
Factor to get: (j + k)(j – k)
Since j and k are both ODD, we know that (j + k)(j – k) = (odd + odd) (odd - odd) = (even)(even) = (2 times some integer)(2 times some integer) = (4)(some integer)

ASIDE: If an integer is even, we can rewrite that number as 2 times some number. For example, we can take 10 and rewrite it as (2)(5)

So, it must be the case that the correct answer must be divisible by 4

Key concept: If integer N is divisible by 4, then the number created by the last 2 digits of N must be divisible by 4.

Check the answer choices.
A) 4818. 18 is NOT divisible by 4. Eliminate A.
B) 5174. 74 is NOT divisible by 4. Eliminate B.
C) 5320. 20 IS divisible by 4. Keep.
D) 5482. 82 is NOT divisible by 4. Eliminate D.
E) 5566. 66 is NOT divisible by 4. Eliminate E.

Answer: C

Cheers,
Brent
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Re: If x and y are odd positive integers, and x and y both have an odd  [#permalink]

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New post 15 Sep 2019, 07:49
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x and y are both odd and both have an odd number of integers; hence, x and y are both perfect perfect square of odd integers.

\(x=(2a+1)^2\), where a is an integer
\(y=(2b+1)^2\), where b is an integer

\(x-y= 4a^2+4a+1-4b^2-4b-1\)\(= 4(a^2+a-b^2-b)\)
hence, value of x-y must be a multiple of 4

Only option C is a multiple of 4

IMO C


GMATPrepNow wrote:
If x and y are positive odd integers, and both numbers have an odd number of positive divisors, which of the following could be the value of x-y?

A) 4818
B) 5174
C) 5320
D) 5482
E) 5566
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If x and y are odd positive integers, and x and y both have an odd  [#permalink]

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New post 15 Sep 2019, 09:33
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GMATPrepNow wrote:
If x and y are positive odd integers, and both numbers have an odd number of positive divisors, which of the following could be the value of x-y?

A) 4818
B) 5174
C) 5320
D) 5482
E) 5566


Given:
1. x and y are positive odd integers
2. Both numbers have an odd number of positive divisors.

Asked: Which of the following could be the value of x-y?

x, y are positive odd integers of the form a^2, b^2 respectively

x - y = a^2 - b^2 = (a+b)(a-b)
Since a & b are odd ; a+b & a-b are even; (a+b)(a-b) is a multiple of 4

70^2 = 4900
71^2 = 5041
72^2 = 5184
73^2 = 5329
74^2 = 5476
75^2 = 5625

A) 4818; NOT a multiple of 4
B) 5174; NOT a multiple of 4
C) 5320; MULTIPLE of 4; 5320 = 5329 - 9 = 73^2 - 3^2 = 76*70
D) 5482; NOT a multiple of 4
E) 5566; NOT a multiple of 4

IMO C
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Re: If x and y are odd positive integers, and x and y both have an odd  [#permalink]

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New post 15 Sep 2019, 09:56
from the given info we know that both x& y are odd integers and are perfect squares as well.
we can test for few values of given condition
3^2-1^2=8
5^2-3^2= 16
5^2-1^2= 24
13^2-11^2 = 48
21^2-11^2= 320
we observe that all the integers are divisible by 8
seeing the answer options
A) 4818 ; has only 1 factor of 2
B) 5174; has only 1 factor of 2
C) 5320; has 3 factors of 2 hence divisible by 8 sufficient
D) 5482; has only 1 factor of 2
E) 5566' has only 1 factor of 2

IMO C

GMATPrepNow wrote:
If x and y are positive odd integers, and both numbers have an odd number of positive divisors, which of the following could be the value of x-y?

A) 4818
B) 5174
C) 5320
D) 5482
E) 5566
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Re: If x and y are odd positive integers, and x and y both have an odd  [#permalink]

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New post 22 Sep 2019, 13:25
This is quite simple,really.When you subtract two odd perfect squares ,you get an even number. To be more specific let's take a look at this :
25-9=(5+3)(5-3)=8 × 2.
Another: 81-25=(9-5)(9+5)=4×14.
What you will realise is that 8×2 and 4×14 must be divisible by at least 4 or 8 since the difference of two squares of two odd numbers is the product of 2 even integers.

Only C is divisible by 4 because its last 2 digits form a number divisible by 4. ANSWER IS C
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Re: If x and y are odd positive integers, and x and y both have an odd  [#permalink]

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New post 08 Nov 2019, 23:04
The clue lies in that the odd integers having odd factors...take 3 for example...it has 2 factors...but 3^2=9 has 3 factors 1,3&9...true for all odd squares...so x and y are perfect squares and have to be odd. Now using the formula a^2-b^2=(a+b)(a-b) and the property odd plus odd is even and odd minus odd is even, we get x-y as the product of 2 even numbers.. so a multiple of 4. ...C is the only number who's last 2 digits are divisibke by 4. Hence C

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Re: If x and y are odd positive integers, and x and y both have an odd   [#permalink] 08 Nov 2019, 23:04
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