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08 Jan 2006, 13:06
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If X,Y r integers, is X*Y divisible by 3 ?
1) (X+Y)^2 divisible by 9
2) (X-Y)^2 divisible by 9
1) (X+Y)^2 divisible by 9
2) (X-Y)^2 divisible by 9
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08 Jan 2006, 20:12
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Is it C?
1) (X+Y)^ is divisible by 9
1) Given (X+Y)^ is divisible by 9.
Hence (X+Y)^2 = 9 * K ---> K, being some constant
=> X+Y = (+/-) 3 SQRT (K)
Since X & Y are integers, it must be true that SQRT (K) is also an integer say K1.
Hence X+Y = (+/-) 3 * K1 --(1)
There are several values that satisfy this eqn: (0, 3), (1, 5), but X*Y may or may not be divisible by 3 ==> INSUFFICIENT.
2) Given (X-Y)^ = 9 * L ---> L, being some constant
=> X-Y = (+/-) 3*L1 -- (2)
There are several values that satisfy this eqn: (4,1), (6,3).....
Similar to (1) this is also INSUFFICIENT
Combining (1) & (2):
If we plug in values for X & Y.
....(-6, -3) (-3, 0) (0, 3) (3, 6) ... are the only types of values that satisfy BOTH (1) and (2)
For all of these X*Y is a multiple of 3. Hence C
1) (X+Y)^ is divisible by 9
1) Given (X+Y)^ is divisible by 9.
Hence (X+Y)^2 = 9 * K ---> K, being some constant
=> X+Y = (+/-) 3 SQRT (K)
Since X & Y are integers, it must be true that SQRT (K) is also an integer say K1.
Hence X+Y = (+/-) 3 * K1 --(1)
There are several values that satisfy this eqn: (0, 3), (1, 5), but X*Y may or may not be divisible by 3 ==> INSUFFICIENT.
2) Given (X-Y)^ = 9 * L ---> L, being some constant
=> X-Y = (+/-) 3*L1 -- (2)
There are several values that satisfy this eqn: (4,1), (6,3).....
Similar to (1) this is also INSUFFICIENT
Combining (1) & (2):
If we plug in values for X & Y.
....(-6, -3) (-3, 0) (0, 3) (3, 6) ... are the only types of values that satisfy BOTH (1) and (2)
For all of these X*Y is a multiple of 3. Hence C
08 Jan 2006, 21:06
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E?
1. (x+y)^2 = 9*b, where b is int. Not suf
2. (x-y)^2-9*c, where c is int. Not suf.
Together. Substruct 2 from 1. 4*x*y=9(b-c)
x*y/3=3*(b-c)/4 should be int in order for x*y to be divisible by 3. But it is not always true. It's true only when b-c is divisible by 4.
1. (x+y)^2 = 9*b, where b is int. Not suf
2. (x-y)^2-9*c, where c is int. Not suf.
Together. Substruct 2 from 1. 4*x*y=9(b-c)
x*y/3=3*(b-c)/4 should be int in order for x*y to be divisible by 3. But it is not always true. It's true only when b-c is divisible by 4.
