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Manager  Joined: 10 Feb 2011
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If b is an integer, is (a^2 + b^2)^(1/2) an integer?  [#permalink]

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If b is an integer, is $$\sqrt {a^2+b^2}$$ an integer?

(1) a^2 + b^2 is an integer.
(2) a^2 – 3b^2 = 0

DS46402.01
Math Expert V
Joined: 02 Sep 2009
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If b is an integer, is (a^2 + b^2)^(1/2) an integer?  [#permalink]

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If b is an integer, is $$\sqrt {a^2+b^2}$$ an integer?

(1) a^2 + b^2 is an integer. Clearly insufficient: if a^2 + b^2 is a perfect square, say 4, then the answer is YES but if a^2 + b^2 is NOT a perfect square, say 5, then the answer is NO. Not sufficient.

(2) a^2 – 3b^2 = 0 --> $$a^2=3b^2$$ --> $$\sqrt{a^2+b^2}=\sqrt{3b^2+b^2}=\sqrt{4b^2}=2|b|=integer$$. Sufficient.

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Re: If b is an integer, is (a^2 + b^2)^(1/2) an integer?  [#permalink]

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(1) Let a be $$\sqrt{3}$$ and b be 2, then $$a^2 + b^2$$ = 3 + 4 = 7 which is an integer, but $$\sqrt{a^2 + b^2}$$= $$\sqrt{7}$$ which is not an integer.
At the same time, if a = 3, b = 4, then $$a^2 + b^2$$ = 25 which is an integer, but $$\sqrt{a^2 + b^2}$$ = 5 which is an integer.

Since we can get a Yes or No for the question, statement is Insufficient.

(2) $$a^2 = 3b^2$$
Substituting this, we get $$\sqrt{3b^2 + b^2}$$ = $$\sqrt{4b^2}$$ = 2b. Since b is an integer, 2b is also an integer. Hence Sufficient.

IMO, Option B
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Re: If b is an integer, is (a^2 + b^2)^(1/2) an integer?  [#permalink]

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nkin wrote:
(1) Let a be $$\sqrt{3}$$ and b be 2, then $$a^2 + b^2$$ = 3 + 4 = 7 which is an integer, but $$\sqrt{a^2 + b^2}$$= $$\sqrt{7}$$ which is not an integer.
At the same time, if a = 3, b = 4, then $$a^2 + b^2$$ = 25 which is an integer, but $$\sqrt{a^2 + b^2}$$ = 5 which is an integer.

Since we can get a Yes or No for the question, statement is Insufficient.

(2) $$a^2 = 3b^2$$
Substituting this, we get $$\sqrt{3b^2 + b^2}$$ = $$\sqrt{4b^2}$$ = 2b. Since b is an integer, 2b is also an integer. Hence Sufficient.

IMO, Option B

Hi nkin, I think didn't get when you Let a be $$\sqrt{3}$$

just double check if I understand your reasoning correctly, so when we plug in these values we get the following ?

$$\sqrt{\sqrt({3})^2+2^2}$$ = $$\sqrt{3+4}$$ = $$\sqrt{7}$$

Do i understand you correctly ? thank you VP  D
Joined: 09 Mar 2016
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Re: If b is an integer, is (a^2 + b^2)^(1/2) an integer?  [#permalink]

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Bunuel wrote:
banksy wrote:
If b is an integer, is sq(a^2+b^2) an integer?
(1) a2 + b2 is an integer.
(2) a^2 – 3b^2 = 0

If b is an integer, is sqrt(a^2+b^2) an integer?

(1) a^2 + b^2 is an integer. Clearly insufficient: if a^2 + b^2 is a perfect square, say 4, then the answer is YES but if a^2 + b^2 is NOT a perfect square, say 5, then the answer is NO. Not sufficient.
(2) a^2 – 3b^2 = 0 --> $$a^2=3b^2$$ --> $$\sqrt{a^2+b^2}=\sqrt{3b^2+b^2}=\sqrt{4b^2}=2|b|=integer$$. Sufficient.

Bunuel can you pls explain how would you get 5 ?
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Re: If b is an integer, is (a^2 + b^2)^(1/2) an integer?  [#permalink]

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dave13 wrote:
nkin wrote:
(1) Let a be $$\sqrt{3}$$ and b be 2, then $$a^2 + b^2$$ = 3 + 4 = 7 which is an integer, but $$\sqrt{a^2 + b^2}$$= $$\sqrt{7}$$ which is not an integer.
At the same time, if a = 3, b = 4, then $$a^2 + b^2$$ = 25 which is an integer, but $$\sqrt{a^2 + b^2}$$ = 5 which is an integer.

Since we can get a Yes or No for the question, statement is Insufficient.

(2) $$a^2 = 3b^2$$
Substituting this, we get $$\sqrt{3b^2 + b^2}$$ = $$\sqrt{4b^2}$$ = 2b. Since b is an integer, 2b is also an integer. Hence Sufficient.

IMO, Option B

Hi nkin, I think didn't get when you Let a be $$\sqrt{3}$$

just double check if I understand your reasoning correctly, so when we plug in these values we get the following ?

$$\sqrt{\sqrt({3})^2+2^2}$$ = $$\sqrt{3+4}$$ = $$\sqrt{7}$$

Do i understand you correctly ? thank you dave13, yes that is correct. Statement 1 says $$a^2 + b^2$$ is an integer, so here $$a^2 + b^2$$ is 7, which is an integer but sqrt(a^2 + b^2) or sqrt(7) is not an integer as asked in the question stem. This is just to give an example where using statement (1), we are not able to answer the question stem definitively.
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WE: Manufacturing and Production (Energy and Utilities)
Re: If b is an integer, is (a^2 + b^2)^(1/2) an integer?  [#permalink]

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1
dave13 wrote:
Bunuel wrote:
banksy wrote:
If b is an integer, is sq(a^2+b^2) an integer?
(1) a2 + b2 is an integer.
(2) a^2 – 3b^2 = 0

If b is an integer, is sqrt(a^2+b^2) an integer?

(1) a^2 + b^2 is an integer. Clearly insufficient: if a^2 + b^2 is a perfect square, say 4, then the answer is YES but if a^2 + b^2 is NOT a perfect square, say 5, then the answer is NO. Not sufficient.
(2) a^2 – 3b^2 = 0 --> $$a^2=3b^2$$ --> $$\sqrt{a^2+b^2}=\sqrt{3b^2+b^2}=\sqrt{4b^2}=2|b|=integer$$. Sufficient.

Bunuel can you pls explain how would you get 5 ?

dave13, Here's what Bunuel is saying: If you would get 5, i.e. if a = 1 and b = 2, that a^2 + b^2 = 1 + 4 = 5, then 5 is an integer as per statement 1, but then again, sqrt (a^2 + b^2) = sqrt(5) is not an integer, as per the question stem's ask.
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Re: If b is an integer, is sqr root (a^2 + b^2) an integer? 1)  [#permalink]

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