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If x and y are positive integers, is x + y an even integer?
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(1) The product of the greatest common factor of x and y and the least common multiple of x and y is even
Let's take examples:
Case 1: (3,6) -> LCM = 6, GCF = 3
product of LCM and GCF = 3x16 = 18 (even)
sum of x and y = 3 + 6 (odd)
Case 2: (2,6) -> LCM = 6, GCF = 2
product of LCM and GCF = 2x6 = 12 (even)
sum of x and y = 2 + 6 = 8 (even)
Clearly statement 1 is not sufficient on it's own
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(2) The greatest common factor of x and y is 1
Let's take examples again.
Case 1: (2,3) -> GCF = 1
x + y = 2 + 3 = 5 (odd)
Case 2: (3,5) -> GCF = 1
x + y = 3 + 5 = 8 (even)
Clearly statement 2 is insufficient
Now, let's take both together.
GCF = 1 implies the numbers are co-primes. Meaning, they can't have any common factor other than 1.
LCM x GCF is even, implies LCM is even.
Now, LCM x GCF = xy (is even)
Since product is even, at least one of the numbers has to be even.
Both the numbers can't even even, because then GCF will be at least 2.
This means only 1 of the numbers if even and the other is odd.
So, their sum will be EVEN + ODD = ODD
=> two statements together were sufficient. Answer: C
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