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#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # If x and y are odd positive integers, and x and y both have an odd

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GMAT Club Legend  V
Joined: 12 Sep 2015
Posts: 4219
If x and y are odd positive integers, and x and y both have an odd  [#permalink]

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13 00:00

Difficulty:   75% (hard)

Question Stats: 51% (02:11) correct 49% (02:22) wrong based on 79 sessions

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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

_________________
GMAT Club Legend  V
Joined: 12 Sep 2015
Posts: 4219
Re: If x and y are odd positive integers, and x and y both have an odd  [#permalink]

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Top Contributor
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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.

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.

Cheers,
Brent
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VP  V
Joined: 19 Oct 2018
Posts: 1293
Location: India
Re: If x and y are odd positive integers, and x and y both have an odd  [#permalink]

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4
7
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
##### General Discussion
SVP  D
Joined: 03 Jun 2019
Posts: 1940
Location: India
GMAT 1: 690 Q50 V34 If x and y are odd positive integers, and x and y both have an odd  [#permalink]

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1
1
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
GMAT Club Legend  V
Joined: 18 Aug 2017
Posts: 5708
Location: India
Concentration: Sustainability, Marketing
GPA: 4
WE: Marketing (Energy and Utilities)
Re: If x and y are odd positive integers, and x and y both have an odd  [#permalink]

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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
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
Manager  G
Joined: 10 Jun 2019
Posts: 119
Re: If x and y are odd positive integers, and x and y both have an odd  [#permalink]

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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
Intern  Joined: 11 Feb 2018
Posts: 4
Re: If x and y are odd positive integers, and x and y both have an odd  [#permalink]

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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

Posted from my mobile device 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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