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

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

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New post 30 Sep 2017, 01:43
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Kudos [?]: 132680 [0], given: 12331

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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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New post 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

Kudos [?]: 5 [0], given: 36

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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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New post 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

Option C

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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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New post 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.

Answer is C

Kudos [?]: 8 [0], given: 8

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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