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

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

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16 May 2012, 17:29
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Which of the following expressions can be written as an integer?

1. $$(\sqrt{82} + \sqrt{82})^2$$

2. $$82*\sqrt{82}$$

3. $$\frac{\sqrt{82}*\sqrt{82}}{82}$$

A. none
B. 1 only
C. 3 only
D. 1 and 2
E. 1 and 3
[Reveal] Spoiler: OA

Last edited by Bunuel on 08 Aug 2017, 03:39, edited 2 times in total.
Edited the question

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

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17 May 2012, 01:05
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eybrj2 wrote:
Which of the following expression can be written as an integer?

1) (\sqrt{82} + \sqrt{82})^2
2) 82*\sqrt{82}
3) \sqrt{82}*\sqrt{82}/82

a. none
b. 1 only
c. 3 only
d. 1 and 2
e. 1 and 3

I doubt OA.

Hi

Option E is correct:

1) (\sqrt{82} + \sqrt{82})^2 = (2*\sqrt{82)^2= 4*82= Integer
2) 82*[square_root]82}= not an integer
3) (\sqrt{82} * \sqrt{82})/82
= 82/82=1= Integer
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Re: Which of the following expressions can be written as an integer? [#permalink]

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17 May 2012, 01:25
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eybrj2 wrote:
Which of the following expression can be written as an integer?

1. $$(\sqrt{82} + \sqrt{82})^2$$

2. $$82*\sqrt{82}$$

3. $$\frac{\sqrt{82}*\sqrt{82}}{82}$$

A. none
B. 1 only
C. 3 only
D. 1 and 2
E. 1 and 3

I doubt OA.

1. $$(\sqrt{82} + \sqrt{82})^2=(2\sqrt{82})^2=4*82=integer$$

2. $$82*\sqrt{82}$$ --> since $$\sqrt{82}$$ is not an integer then $$82*\sqrt{82}$$ is not integer either.

3. $$\frac{\sqrt{82}*\sqrt{82}}{82}=\frac{82}{82}=1=integer$$.

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

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10 Nov 2016, 18:33
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Re: Which of the following expressions can be written as an integer? [#permalink]

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08 Aug 2017, 03:31
Bunuel wrote:
eybrj2 wrote:
Which of the following expression can be written as an integer?

1. $$(\sqrt{82} + \sqrt{82})^2$$

2. $$82*\sqrt{82}$$

3. $$\frac{\sqrt{82}*\sqrt{82}}{82}$$

A. none
B. 1 only
C. 3 only
D. 1 and 2
E. 1 and 3

I doubt OA.

1. $$(\sqrt{82} + \sqrt{82})^2=(2\sqrt{82})^2=4*82=integer$$

2. $$82*\sqrt{82}$$ --> since $$\sqrt{82}$$ is not an integer then $$82*\sqrt{82}$$ is not integer either.

3. $$\frac{\sqrt{82}*\sqrt{82}}{82}=\frac{82}{82}=1=integer$$.

Hello Bunuel

Your explanation is to the point but in the official explanation, its mentioned:

82* 82^3/2

82^3= 2^3 *41^3

How did we get this cube?

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

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08 Aug 2017, 03:42
Shiv2016 wrote:
Bunuel wrote:
eybrj2 wrote:
Which of the following expression can be written as an integer?

1. $$(\sqrt{82} + \sqrt{82})^2$$

2. $$82*\sqrt{82}$$

3. $$\frac{\sqrt{82}*\sqrt{82}}{82}$$

A. none
B. 1 only
C. 3 only
D. 1 and 2
E. 1 and 3

I doubt OA.

1. $$(\sqrt{82} + \sqrt{82})^2=(2\sqrt{82})^2=4*82=integer$$

2. $$82*\sqrt{82}$$ --> since $$\sqrt{82}$$ is not an integer then $$82*\sqrt{82}$$ is not integer either.

3. $$\frac{\sqrt{82}*\sqrt{82}}{82}=\frac{82}{82}=1=integer$$.

Hello Bunuel

Your explanation is to the point but in the official explanation, its mentioned:

82* 82^3/2

82^3= 2^3 *41^3

How did we get this cube?

Could you please post a screenshot or post entire solution?
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Which of the following expressions can be written as an integer? [#permalink]

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08 Aug 2017, 03:50
Expression II does not represent an integer because (82)* 82^1/2= 82^3/2 and 82^3= 2^3* 41^3 is not a perfect square. Regarding this last assertion, note that the square of any integer has the property that each of its distinct prime factors is repeated an even number of times.

(Explanation taken from official guide 2017)
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Re: Which of the following expressions can be written as an integer? [#permalink]

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08 Aug 2017, 04:00
Shiv2016 wrote:
Expression II does not represent an integer because (82)* 82^1/2= 82^3/2 and 82^3= 2^3* 41^3 is not a perfect square. Regarding this last assertion, note that the square of any integer has the property that each of its distinct prime factors is repeated an even number of times.

(Explanation taken from official guide 2017)

$$82*82^{(\frac{1}{2})}=82^{(1+\frac{1}{2})}=82^{\frac{3}{2}}=\sqrt{82^3}$$
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Re: Which of the following expressions can be written as an integer? [#permalink]

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10 Aug 2017, 10:48
eybrj2 wrote:
Which of the following expressions can be written as an integer?

1. $$(\sqrt{82} + \sqrt{82})^2$$

2. $$82*\sqrt{82}$$

3. $$\frac{\sqrt{82}*\sqrt{82}}{82}$$

A. none
B. 1 only
C. 3 only
D. 1 and 2
E. 1 and 3

Let’s simplify each expression:

1. (√82 + √82)^2

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

We see that this an integer.

2. 82√82

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

3. (√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] 10 Aug 2017, 10:48
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