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# If a and b are positive integers, is a^2 + b^2 divisible by 5?

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Math Expert
Joined: 02 Sep 2009
Posts: 47037
If a and b are positive integers, is a^2 + b^2 divisible by 5? [#permalink]

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30 Sep 2017, 01:43
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57% (01:17) correct 43% (01:00) wrong based on 61 sessions

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If $$a$$ and $$b$$ are positive integers, is $$a^2 + b^2$$ divisible by 5?

(1) $$2ab$$ is divisible by 5

(2) $$a - b$$ is divisible by 5

M19-11

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Joined: 14 Sep 2016
Posts: 149
Re: If a and b are positive integers, is a^2 + b^2 divisible by 5? [#permalink]

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30 Sep 2017, 02:46
Bunuel wrote:
If $$a$$ and $$b$$ are positive integers, is $$a^2 + b^2$$ divisible by 5?

(1) $$2ab$$ is divisible by 5

(2) $$a - b$$ is divisible by 5

M19-11

Statement (1) says tat 2ab is divisible by 5, hence option is insufficient.
Statement (2) Doesnt say anything about the value of a & b hence insufficient

taking both options together we get : (a-d)^2 = 25 => a^2 + b^2 - 2ab = 25

=> a^2 + b^2 = 25 + 2ab

since both 25 and 2ab is divisible by 5 we can conclude that a^2 + b^2 is divisible by 5
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Re: If a and b are positive integers, is a^2 + b^2 divisible by 5? [#permalink]

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30 Sep 2017, 02:47
Bunuel wrote:
If $$a$$ and $$b$$ are positive integers, is $$a^2 + b^2$$ divisible by 5?

(1) $$2ab$$ is divisible by 5

(2) $$a - b$$ is divisible by 5

M19-11

Statement 1: if $$a=5$$ and $$b=1$$, then $$2ab$$ is divisible by $$5$$ but $$a^2+b^2$$ is not, where as if $$a=b=5$$, then both $$2ab$$ and $$a^2+b^2$$ is divisible by $$5$$. Hence Insufficient

Statement 2: if $$a=b=5$$, then both $$a-b$$ & $$a^2+b^2$$ is divisible by $$5$$, but if $$a=6$$ and $$b=1$$, then $$a-b$$ is divisible by $$5$$ but $$a^2+b^2$$ is not. Hence Insufficient

Combining 1 & 2, we can write $$a^2+b^2=(a-b)^2+2ab$$, Now $$a-b$$ is divisible by $$5$$ hence its square will also be divisible by $$5$$ and $$2ab$$ is divisible by $$5$$. So we can say that $$a^2+b^2$$ is divisible by $$5$$. Hence Sufficient

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Re: If a and b are positive integers, is a^2 + b^2 divisible by 5? [#permalink]

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30 Sep 2017, 05:25
Bunuel wrote:
If $$a$$ and $$b$$ are positive integers, is $$a^2 + b^2$$ divisible by 5?

(1) $$2ab$$ is divisible by 5

(2) $$a - b$$ is divisible by 5

M19-11

Yes no type Qs, a
Given info- a and b are positive integer then a2+b2/5 =integer??

St-1 2ab is divisible by 5, then a is multiple of 5 or b is multiple of 5 or both a & b is multiple of 5.
If a& b both multiple of 5 then sufficient if a only or b only is multiple then insufficient.

St-2 a-b is multiple of 5, a and b both multiple of 5 then sufficient also a and b can not be multiple of 5 eg. 8-3 is divisible by 5. insufficient.

Combine--If either a or b is multiple of 5 then other one has to be multiple of 5 to satisfy a-b is multiple of 5. then sufficient.
if a is 10 and b is 2 then a-b can not be divisible by 5. for both statement true and a and b has to be multiple of 5.

Re: If a and b are positive integers, is a^2 + b^2 divisible by 5?   [#permalink] 30 Sep 2017, 05:25
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