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Math Expert V
Joined: 02 Sep 2009
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Which of the following expressions can be written as an integer?  [#permalink]

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11 00:00

Difficulty:   5% (low)

Question Stats: 79% (00:43) correct 21% (00:44) wrong based on 1062 sessions

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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]

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2
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

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Re: Which of the following expressions can be written as an integer?  [#permalink]

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2
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]

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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

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Posts: 11
Re: Which of the following expressions can be written as an integer?  [#permalink]

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1
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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GMAT 1: 730 Q48 V42 Re: Which of the following expressions can be written as an integer?  [#permalink]

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1
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]

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4
Attached is a visual that should help.
Attachments Screen Shot 2016-05-10 at 5.49.00 PM.png [ 153.51 KiB | Viewed 11982 times ]

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GRE 1: Q169 V154 Re: Which of the following expressions can be written as an integer?  [#permalink]

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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]

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Top Contributor
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.

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GMAT 1: 410 Q18 V27 GMAT 2: 490 Q35 V23 Re: Which of the following expressions can be written as an integer?  [#permalink]

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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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Re: Which of the following expressions can be written as an integer?  [#permalink]

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1
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.

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Re: Which of the following expressions can be written as an integer?  [#permalink]

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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

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 Re: Which of the following expressions can be written as an integer?   [#permalink] 14 Sep 2019, 00:04
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