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Re: HOT Competition 1 Sep/8PM: If x and y are positive integers, is x + y
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01 Sep 2020, 21:26
IMO C
Note: HCF (x,y) * LCM(x,y) = Product of two numbers = x*y
If x and y are positive integers, is x + y an even integer?
(1) The product of the greatest common factor of x and y and the least common multiple of x and y is even
HCF (x,y) * LCM(x,y) = 2K (for some non negative integer K)
x * y= 2K
Many cases possible:
eg: x=1, y=2 , then HCF=1, LCM=2 & (x+y)=3
if, x=2, y=4, then HCF=2, LCM=4 & (x+y)=6
Since , (x+y) can be both Even and Odd,
Not Sufficient
(2) The greatest common factor of x and y is 1
HCF (x,y) = 1
This means x & y are co-prime.
Since no other information is provided, here also multiple cases possible.
eg: x=3, y=5, HCF=1, (x+y)=8
if x=2, y=5, HCF=1, (x+y)=7
Since , (x+y) can be both Even and Odd,
Not Sufficient
Together:
HCF (x,y) = 1
This means x & y are co-prime.
HCF (x,y) * LCM(x,y) = 2K
=> LCM (x,y)= 2K (K positive integer)
Let x=pa & y=pb (HCF= p, LCM=pab, where a & b coprime)
pab= 2K
=> ab = 2k
Note: Two integers product is even when (Even x Even) 0r (OddxEven)
But if Even x Even Case, HCF min will be 2, Which is not possible as per information provided.
So, ab= Odd x Even => One of x,y is even & other odd.
Therefore, (x+y)= (odd+even)=0dd
So, Sufficient
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