HOT Competition 1 Sep/8PM: If x and y are positive integers, is x + y

Question

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

(2) The greatest common factor of x and y is 1

M36-55

Bunuel · Accepted Answer

Official Solution:

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
 
 The product of the GCF and LCM of two positive integers is equal to the product of those integers . So, we are given that  GCF(x,y)*LCM(x,y)=xy=even . Now,  xy=even  means that at least one of the unknowns is even. If both are even, then  x + y=even  but if only one of them is even and another is odd, then  x + y=odd . For example, consider  x=2  and  y=4  for an YES answer and  x=2  and  y=3  for a NO answer. Not sufficient. 
 
(2) The greatest common factor of  x  and  y  is 1
 
Clearly insufficient. For example, consider  x=3  and  y=1  for an YES answer and  x=3  and  y=2  for a NO answer. Not sufficient.
 
(1)+(2) From (1) we got that either both unknowns are even or one is even and another is odd. The first case is ruled out by (2), since if both are even, then they both would share 2 as a common factor, while (2) says that the greatest common factor of  x  and  y  is 1. Thus, one of the unknowns is even and another is odd:  x + y=odd . Sufficient.

Answer: C

binsnitin11 · Answer

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
product of GCF(x,y) and LCM (X,y) is even
GCF(x,y)*LCM (x,y) = x*y
what if x=3 and y =6 GCF(x,y)*LCM (x,y) = even
x+y = odd
what if x=2 and y =4 GCF(x,y)*LCM (x,y) = even
x+y = even

Not Sufficient

(2) The greatest common factor of x and y is 1
GCF (x,y) = 1, so either both are odd or one is even and one is odd
x=2, y = 7 GCF (2,7) = 1 
2+7 = odd
x=3 y = 7 GCF(3,7) = 1
3+7 = even

Not Sufficient

Combining 1 and 2 
we get to know that one is odd and one is even
odd + even = odd
sufficient

IMO C

Tanushka77 · Answer

If x and y are positive integers, is x + y an even integer?

It is possible when either both are even or both are odd

(1) The product of the greatest common factor of x and y and the least common multiple of x and y is even

This implies that x*y=even. However, they could both be even, or even and odd. 
Not sufficient

(2) The greatest common factor of x and y is 1

Examples:
Two odds - X=9,y=11 then yes
Odd and even - X=1, y=2 then no
Not suff. 
Two evens - impossible

Taking together, 
X*y=even and thus we can take the only possible option from the second. It is - odd and even
Thus, thus is sufficient

The answer is C

Posted from my mobile device

michael30 · Answer

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

(2) The greatest common factor of x and y is 1

Solution:

Statement 1: Insufficient
Test cases: 
a. x = 2, y = 3
 GCF = 1
 LCM = 6

x + y = 5 (odd)

b. x = 4, y = 8
 GCF = 4
 LCM = 8

x + y = 12 (even)

Statement 2: Insufficient
Test cases: 
a. x = 2, y = 3
 GCF = 1

x + y = 5 (odd)

b. x = 5, y = 1
 GCF = 1
 
 x + y = 6 (even)

Together, since GCF = 1, x + y = odd number based on test case in Statement 1.

The answer is C.

Sushil117 · Answer

(1) The product of the greatest common factor of x and y and the least common multiple of x and y is even
Means either LCM or HCf is positive. Not sufficient 
(2) The greatest common factor of x and y is 1
Means numbers are prime. Still sufficient

However, together it can be understood either x or y is 2,to make LCM even and other is odd

IMO c is the answer

Posted from my mobile device

123456789007 · Answer

from option 1
we know that HCF*LCM=Product of the two numbers even*even= even even*odd=even not sufficient
from 2 odd+odd=even or odd+ even=odd not sufficient
combining both is sufficient 
c

NiftyNiffler · Answer

If x and y are positive integers, is x + y an even integer?

st1) This means the multiplication of 2 integers is even (since LCMxHCF = xy), but that could mean either x & y both are even or one of them is. INSUFFICIENT

st2) LCM = xy (the numbers are co-primes, e.g., x = 3, y = 5 or x = 2, y = 3) INSUFFICIENT

Both taken together, xy = LCM = even , in this case both x & y can't be EVEN because in that case, minimum GCF will be 2. So, One of them is EVEN & the other ODD.

SUFFICIENT

The answer should be  C

debjit1990 · Answer

x,y=+ve int
is x+y=even?
we know, even+even=even or
 odd+odd=even

(1) The product of the greatest common factor of x and y and the least common multiple of x and y is even
take x=2 y=4
gcd=2 & lcm=4..product=2*4=8=even
so both even..x+y=even..yes
take x=2,y=3
gcd=1 & lcm=6..product=6=even
so one even 7 another odd..x+y=odd..no
not sufficient

(2) The greatest common factor of x and y is 1
take x=3 & y=8 gcd=1..x+y=11 odd no
take x=3 & y=5 gcd=1..x+y=8 even yes
not sufficient

combine,
gcd=1 & gcd*lcm=even
so, we cannot take both x,y =odd or even
so we have to take one odd & another even
we know, odd+even=odd no
sufficien..ans C

RajatJ79 · Answer

If x and y are positive integers, is x + y an even integer?

Pre-thinking :  x+y will be even if both intigers are even only or odd only.

Solution :

(1) The product of the greatest common factor of x and y and the least common multiple of x and y is even

Considering statement (1)
Product of gcf and lcm of two numbers is = product of numbers themselves.
Thus, x*y = even integer.
However, product of two integers is even if even one of the integers is even.
Thus, x and y could be 1 and 2 or 4 and 6. Using above samples, x+y could be even or also odd.

As there is no unique answer,  statement (1) by itself is not sufficient.

(2) The greatest common factor of x and y is 1

Considering statement (2)
The gcf of x and y can be 1 for only two cases, any integer and 1 also two prime integers.
Thus, x and y could be 1 and 2 or 5 and 7. Using above samples, x+y could be even or also odd.

As there is no unique answer,  statement (2) by itself is not sufficient.

Combining statement (1) and statement (2) 
If gcf*lcm i.e. x*y= even and gcf=1, x and y can only be 1 and 2 from the samples considered in the two statements individually.
Both statements combined are enough.

Answer should be Option C.

Posted from my mobile device

Thekingmaker · Answer

it's pretty simple we have to figure out the different possiblities '
A- for example let's take 2,3 , gc=1, lcm= 6 , together 2+3= 5 odd n0 , 2,4 , gc= 2 , together 6 even yes ,clearly not suff
B - same examples 2,3= odd , 5,7 ,gc= 1, lcm= 35 , together 12 even clearly each individually is insuff 
when combined it;s sufficient for instance it;s odd whenever the gcd is 1 and lcm is even 2,3 or 2,5

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HOT Competition 1 Sep/8PM: If x and y are positive integers, is x + y

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HOT Competition 1 Sep/8PM: If x and y are positive integers, is x + y

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