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Which of the following expressions can be written as an integer?

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Bunuel
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Which of the following expressions can be written as an integer? [#permalink] New post   16 Oct 2015, 07:27
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Which of the following expressions can be written as an integer?

I. \((\sqrt{82} + \sqrt{82})^2\)

II. \(82\sqrt{82}\)

III. \(\frac{\sqrt{82}\sqrt{82}}{82}\)

(A) None
(B) I only
(C) III only
(D) I and II
(E) I and III


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Re: Which of the following expressions can be written as an integer? [#permalink] New post   16 Oct 2015, 09:35
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Bunuel wrote:
Which of the following expressions can be written as an integer?

I. \((\sqrt{82} + \sqrt{82})^2\)

II. \(82\sqrt{82}\)

III. \(\frac{\sqrt{82}\sqrt{82}}{82}\)

(A) None
(B) I only
(C) III only
(D) I and II
(E) I and III


I. \((\sqrt{82} + \sqrt{82})^2\)
= \((2\sqrt{82})^2\)
= 4 * 82 = 328 integer

2. \(82\sqrt{82}\)
cannot be expressed as integer

3. \(\frac{\sqrt{82}\sqrt{82}}{82}\)
= 82/82
= 1, which is integer

Answer option E
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Re: Which of the following expressions can be written as an integer? [#permalink] New post   10 May 2016, 17:57
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Attached is a visual that should help.
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Screen Shot 2016-05-10 at 5.49.00 PM.png
Screen Shot 2016-05-10 at 5.49.00 PM.png [ 153.51 KiB | Viewed 30330 times ]

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Re: Which of the following expressions can be written as an integer? [#permalink] New post   16 Oct 2015, 07:38
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1. ( (82)^(1/2) + (82)^(1/2) )^2
= ( 2 *(82)^(1/2))^2
= 4 * 82
= 328
Integer

2. 82 (82)^(1/2)
can't be written as an integer as 82 is not a perfect square .

3. ((82)^(1/2) * (82)^(1/2) )/82
= 1
Integer

Answer E
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Re: Which of the following expressions can be written as an integer? [#permalink] New post   16 Oct 2015, 08:47
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1st:

[SQRT(82) + SQRT(82)]^2

we have formula (a+b)^2 = a^2 +b^2 +2ab
we can rewrite the first statement as 82+82 + 2*[SQRT(82)*SQRT(82)]
[SQRT(82)*SQRT(82)] this is equal to 82. we get an integer.

We can eliminate
(A) None
(C) III only

Statement II:
82*sqrt(82)
we know that 82 is not a perfect square, thus, it is not an integer. Statement II will not yield an integer.
We can eliminate:

(D) I and II


Statement III:
[sqrt(82) * sqrt (82)]/82
sqrt(82) multiplied by itself will result in 82. 82 divide by 82, and get 1. This is an integer.

Eliminate answer choice B, and the only answer choice left, the correct one is E.
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Re: Which of the following expressions can be written as an integer? [#permalink] New post   16 Oct 2015, 12:07
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l can be written as (2√82)^2 ..a integer
lll will equal to 1 so Option E is correct answer.
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Re: Which of the following expressions can be written as an integer? [#permalink] New post   17 Oct 2015, 11:56
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I. We use the identity (a+b)^2 = a^2+2ab+b^2 which gives us:
82 + (2*\(\sqrt{82}\)*\(\sqrt{82}\)) + 82
Since \(\sqrt{82}\)*\(\sqrt{82}\) equals 82 this will yield an integer.

II. No since \(\sqrt{82}\) is not a perfect square and you cannot separate it either into to two perfect squares.


III. Since \(\sqrt{82}\)*\(\sqrt{82}\) equals 82 this will yield an integer : 82/82=1


Answer E : I and III only
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Re: Which of the following expressions can be written as an integer? [#permalink] New post   11 Nov 2016, 14:00
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lets see the options =>
Option 1=> (2√82)^2=> 4*82 => integer
Option 2 => 82√82 => non integer
Option 3 => √82*√82 / 82 => 82/82=> 1 , which is off course an integer.
Hence E
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Re: Which of the following expressions can be written as an integer? [#permalink] New post   23 Mar 2017, 23:16
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i. \({ \left( \sqrt { 82 } +\sqrt { 82 } \right) }^{ 2 }=\left( 2\sqrt { 82 } \right) ^{ 2 }\Rightarrow\) At this moment you already know you have an integer without further calculations.

ii. \(82\sqrt { 82 } \Rightarrow\) We also know that perfect squares don't end in 2, 3, 7 or 8. So, rule this one out as this product won't yield an integer.

iii. \(\cfrac { \sqrt { 82 } \sqrt { 82 } }{ 82 } =\quad \cfrac { \sqrt { 82\ast 82 } }{ 82 } =\cfrac { \sqrt { { 2 }^{ 2 }\ast { 41 }^{ 2 } } }{ 2\ast { 41 } } =\cfrac { 2\ast 41 }{ 2\ast 41 } =\quad 1\quad \Rightarrow \quad integer\)

OA
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E
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Re: Which of the following expressions can be written as an integer? [#permalink] New post   27 Mar 2017, 17:26
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Bunuel wrote:
Which of the following expressions can be written as an integer?

I. \((\sqrt{82} + \sqrt{82})^2\)

II. \(82\sqrt{82}\)

III. \(\frac{\sqrt{82}\sqrt{82}}{82}\)

(A) None
(B) I only
(C) III only
(D) I and II
(E) I and III



Let’s simplify each Roman numeral:

I. (√82 + √82)^2

(√82 + √82)^2 = (2√82)^2 = 4 x 82 = 328

We see that this an integer.

II. 82√82

Since √82 is a non-terminating decimal, 82√82 is not an integer.

III. (√82√82)/82

(√82√82)/82 = 82/82 = 1

We see that this is an integer.

Answer: E
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Which of the following expressions can be written as an integer? [#permalink] New post   Updated on: 08 Feb 2021, 09:14
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Bunuel wrote:
Which of the following expressions can be written as an integer?

I. \((\sqrt{82} + \sqrt{82})^2\)

II. \(82\sqrt{82}\)

III. \(\frac{\sqrt{82}\sqrt{82}}{82}\)

(A) None
(B) I only
(C) III only
(D) I and II
(E) I and III


We'll use the fact that (√82)(√82) = 82

I. (√82 + √82)² = (2√82)² = (2√82)(2√82) = (2)(2)(√82)(√82) = (4)(82) = some integer
II. Doesn't look like it could be an integer. I'm not sure, so I'll move onto to statement III
III. (√82)(√82)/82 = 82/82 = 1

NOTE: As we check each statement, we should also check the answer choices.
Since I and III both work, the correct answer must be E, since there's no option for all 3 to be true.

Answer: E

Originally posted by BrentGMATPrepNow on 11 Nov 2016, 14:59.
Last edited by BrentGMATPrepNow on 08 Feb 2021, 09:14, edited 2 times in total.
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Re: Which of the following expressions can be written as an integer? [#permalink] New post   14 Sep 2019, 00:04
Bunuel wrote:
Which of the following expressions can be written as an integer?

I. \((\sqrt{82} + \sqrt{82})^2\)

II. \(82\sqrt{82}\)

III. \(\frac{\sqrt{82}\sqrt{82}}{82}\)

(A) None
(B) I only
(C) III only
(D) I and II
(E) I and III


Kudos for a correct solution.


Which of the following expressions can be written as an integer?

I. \((\sqrt{82} + \sqrt{82})^2\)
82 + 82 + 2*82 = 4*82 = 328
INTEGER

II. \(82\sqrt{82}\)
82 * \sqrt{82}
9<\sqrt{82}<10 and is NOT an integer
NOT AN INTEGER

III. \(\frac{\sqrt{82}\sqrt{82}}{82}\)
82/82 = 1
INTEGER

IMO E
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Re: Which of the following expressions can be written as an integer? [#permalink] New post   24 Apr 2021, 04:25
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Solution:

We can easily and immediately observe III to be = 82/82 =1 (Eliminate A,B,D)

I - (√82+√82)^2 = (2√82)^2

= 4 X 82 =An integer (eliminate C)

(option e)
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Re: Which of the following expressions can be written as an integer? [#permalink] New post   12 May 2021, 00:52
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Bunuel wrote:
Which of the following expressions can be written as an integer?

I. \((\sqrt{82} + \sqrt{82})^2\)

II. \(82\sqrt{82}\)

III. \(\frac{\sqrt{82}\sqrt{82}}{82}\)

(A) None
(B) I only
(C) III only
(D) I and II
(E) I and III


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