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m^2+n^2=5k+b ; b=0? I. m-n=5z+1 II. m+n=5t+3

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m^2+n^2=5k+b ; b=0? I. m-n=5z+1 II. m+n=5t+3 [#permalink] New post 19 Nov 2008, 12:54
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A
B
C
D
E

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m^2+n^2=5k+b; b=0?

I. m-n=5z+1
II. m+n=5t+3
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Re: DS: divisibility [#permalink] New post 19 Nov 2008, 17:26
this one puzzles me...but I'd pick C

Stmt 1 or 2 by themselves not sufficient..coz you don't know the value of 2mn, when you square both of the LHSs.

Squaring both and adding them results in
m^2+n^2-2mn = 5z + 1
m^2+n^2+2mn = 5t + 3
OR
2m^2+2n^2 = 5(z+t) + 4

simplifying, we get
m^2+n^2 = 5(z+t)/2 + 2

Looks like we can conclude if b = 0 or not from that. Hence C.
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Re: DS: divisibility [#permalink] New post 20 Nov 2008, 00:56
C.
Masuhari , ur approach is right, but u forgot to square right side of the equation.
IF U had done so, u would have come to b=0.
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Re: DS: divisibility [#permalink] New post 20 Nov 2008, 03:01
masuhari wrote:
this one puzzles me...but I'd pick C

Stmt 1 or 2 by themselves not sufficient..coz you don't know the value of 2mn, when you square both of the LHSs.

Squaring both and adding them results in
m^2+n^2-2mn = 5z + 1
m^2+n^2+2mn = 5t + 3
OR
2m^2+2n^2 = 5(z+t) + 4

simplifying, we get
m^2+n^2 = 5(z+t)/2 + 2

Looks like we can conclude if b = 0 or not from that. Hence C.

masuhari, if you correctly square RHS, eventually we will have (25/2)(z^2+t^2) as one of the expressions. How can one know whether this expression is divisible by 5? To be able to say "Yes" or "No" we would need to know whether z^2+t^2 is even.
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Re: DS: divisibility [#permalink] New post 20 Nov 2008, 03:31
I will simply mark E as we do not know whether these variables are integer or fraction.
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Re: DS: divisibility [#permalink] New post 20 Nov 2008, 09:13
scthakur wrote:
I will simply mark E as we do not know whether these variables are integer or fraction.

OA is not E though :?
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Re: DS: divisibility [#permalink] New post 21 Nov 2008, 01:23
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From:
m-n=5z+1
m+n=5t+3
we have : 2m=5(z+t)+4 then (z+t) is divisible by 2
2n=5(t-z)+3 then (t-z) is divisible by 2
and then z,t is divisible by 2 or z=2p;t=2q
Otherwise, from:
m-n=5z+1
m+n=5t=3
we have 2(m^2+n^2)=25(z^2+t^2)+10z+30t+10
or m^2+n^2=25(z^2+t^2)/2+5z+15t+5
we can see that 25(z^2+t^2)/2=25(4p^2+4q^2)/2 is an integer
so (m^2+n^2)=5.land we can conclude that b=0
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Re: DS: divisibility [#permalink] New post 21 Nov 2008, 04:36
ngotuan wrote:
From:
m-n=5z+1
m+n=5t+3
we have : 2m=5(z+t)+4 then (z+t) is divisible by 2
2n=5(t-z)+3 then (t-z) is divisible by 2
and then z,t is divisible by 2 or z=2p;t=2q
Otherwise, from:
m-n=5z+1
m+n=5t=3
we have 2(m^2+n^2)=25(z^2+t^2)+10z+30t+10
or m^2+n^2=25(z^2+t^2)/2+5z+15t+5
we can see that 25(z^2+t^2)/2=25(4p^2+4q^2)/2 is an integer
so (m^2+n^2)=5.land we can conclude that b=0


I have one doubt ngotuan,
t+z is divisible by 2
t-z is divisible by 2
but it does not mean..that t is divisible by 2.
for example, take t=5, z=9
t+z=14 divisible by 2
and t-z=4 divisible by 2
but t & z are not divisible by 2.
....
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Re: DS: divisibility [#permalink] New post 22 Nov 2008, 18:33
oops,sorry.I made a mistake.
We can resolve this :
m^2+n^2=25(z^2+t^2)/2+5z+15t+5
we need to sure that (z^2+t^2)is divisible by 2.
as I mentioned, (t-z) and (t+z) are divisible by 2 , so (t-z)(t+z) is divisible by 2.
We have :
t^2-z^2 is divisible by 2.
Otherwise
(z^2+t^2)=(t^2-z^2)+2z^2 is divisible by 2---> from this we have (m^2+n^2)=5.l
Sorry again!
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Re: DS: divisibility [#permalink] New post 22 Nov 2008, 21:53
ngotuan wrote:
oops,sorry.I made a mistake.
We can resolve this :
m^2+n^2=25(z^2+t^2)/2+5z+15t+5
we need to sure that (z^2+t^2)is divisible by 2.
as I mentioned, (t-z) and (t+z) are divisible by 2 , so (t-z)(t+z) is divisible by 2.
We have :
t^2-z^2 is divisible by 2.
Otherwise
(z^2+t^2)=(t^2-z^2)+2z^2 is divisible by 2---> from this we have (m^2+n^2)=5.l
Sorry again!



Why (t-z) or (t+z) has to be divisible by 2? & How are they divisible by 2?

ok, 2m = 5(t+z) + 4 but do not know that m or n is an integer.
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Re: DS: divisibility [#permalink] New post 30 Nov 2008, 11:23
Guys, OA C. Thanks for a try.

I have just realised that for 2(m^2+n^2)=25(z^2+t^2)+10z+30t+10 to make sense z^2+t^2 needs to be even, otherwise RHS will be odd which would be incorrect given LHS is even.
Re: DS: divisibility   [#permalink] 30 Nov 2008, 11:23
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m^2+n^2=5k+b ; b=0? I. m-n=5z+1 II. m+n=5t+3

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