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


Stat1- The product of the greatest common factor of x and y and the least common multiple of x and y is even. (x,y) can be (1,2) and (2,2) then x+y can be odd and even. So, not sufficient.

Stat2- The greatest common factor of x and y is 1. (x,y) can be (1,2) and (3,5) then x+y can be odd and even. So, not sufficient.

Combining both, we have (x,y), where one term will be odd and another one even. which is sufficient to ans.

So, I think Option C.

We need to know whether x+y is even or odd.
Lets review each statement alone:

(1) The product of the greatest common factor of x and y and the least common multiple of x and y is even
(THIS MEANS THAT THE PRODUCT OF X AND Y IS EVEN BUT WE DONT KNOW ANYTHING ABOUT X NOR Y) STATEMENT 1 IS INSUFFICIENT

(2) The greatest common factor of x and y is 1 (THIS MEANS THAT X AND Y HAVE ONLY 1 AS COMMON FACTOR BUT WE STILL DONT KNOW ANYTHING ABOUT X OR Y. THE RESULT OF THE SUM OF X AND Y COULD BE EVEN OR ODD. FOR INSTANCE, X=2 AND Y=3 THUS THE SUM IS ODD OR X=3 AND Y=7 THUS THE SUM IS EVEN. STATEMENT 2 IS INSUFFICIENT)

If we combine both statements we will know that the product would be even therefore at least 1 of x and y must be even. However, if both were even, the greatest common factor would be even as well. Therefore one of x and y is even and the other one is odd. Thus the sum is odd.

THEREFORE THE ANSWER IS C

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Answer choice C.

i) GCF x LCM = Even

We know that GCF(x,y) x LCM(x,y) = x*y

x*y = even

three possible escenarios:
x = even, y= even..... so x+y = even
x = even, y= odd...... so x+y = odd
x = odd, y= even...... so x+y= odd.

insufficient.

ii) GCF(x,y) =1

so, x and y could be different prime numbers.

x= 2, y=3 ..... x+y = odd
x= 3, y=5 ..... x+y= even

insufficient

i & ii)

To satisfy both conditions, x or y must be even and the other must be odd.

so x+y = odd.
Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

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1. GCF is even & LCM is also even. Hence both the numbers are even. So X+Y is even. Sufficient
2. GCF is 1. This can be true when X=2 & Y=5 or X=3 & Y=5. So we can't determine a definite outcome & hence this option is insufficient.

A is the answer.
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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

From Statement 1,

From this statement it can be inferred that xy is even, so either x or y is even or both x and y are even;

so x + y can be even or odd;

Not Sufficient

From Statement 2,

the greatest common factor = 1 means the numbers do not share any common factor;

Numbers can be 3 & 5 , x + y = even , 2 & 3 , x + y = odd or 2 & 9 , x + y = odd.

Not sufficient.

Combining both,

The numbers do not share a common factor, so if one number has a factor 2, the other number will not have a factor of 2 ( Since the product of LCM and GCD is even, or x*y is even, either x or y is even )

So, one number is odd and the other number is even

And the sum of x and y will be odd;

So, this is sufficient to answer the question

IMO C is the correct answer

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Is x + y = even ?

(1) GCF(x,y) * LCM(x,y) = x*y
Simply put, the greatest common factor of x and y when multiplied to the least common multiple of x and y is equal to the product of x and y.
So, we know that x*y = even. This can happen when at least one of x and y is even: even*even = even or even*odd = even.
However, even + odd = odd, whereas even + even = even.
Insufficient


(2) x and y are co-prime.
If x and y are 2 and 3, then their sum is odd
If x and y are 3 and 5, then their sum is even.
Insufficient


(1)+(2)
If x*y = even, then only one of them has 2 as a factor because x and y are co-prime. So, one of them is even and the second is odd. Even + odd = odd.


Hence C

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For x+y to be an even integer, both x and y should be odd or even .

Statement 1:

we know that x*y=gcd(x,y)*lcm(x,y)=even

so x and y can be both even or either of a or b is even.
so we have three cases:

x=even and y=even(x+y=even)
x=even and y=odd(x+y=odd)
x=odd and y=even(x+y=odd)

So eliminate statement 1.

Statement 2:

GCD of x and y=1

so this means x and y are both co-prime.

x=2 and y=3(gcd=1 and x+y=odd)
x=3 and y=5(gcd=1 and x+y=even)

we have two different result so eliminate statement 2.

Now statement 1+2 tell us that x=even and y=odd or x=odd and y=even.

In either cases sum of x and y will odd.

IMO: C

St1
HCF * LCM = x*y = Even
x = E and y = O
x+y = Odd
Not sufficient
St 2
x= 2 y =3
x+y = 5 Not satisfying
x=3 y = 5
x+y = 8 Satisying
Insufficient

Combining St1 and St2
At least one of x and y will be even
Since HCF is 1, so other must be odd.
Hence sum must be Odd
Sufficient
Ans C

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I am considering 3 cases

1) 1+2 = 3 i.e Odd + Even
2) 2+4 = 6 i.e Even + Even
3) 5+7 = 12 i.e Odd + Odd

With Statement 1 :

Test cases 1 and 2 will get satisfied.So it's not sufficient

With Statement 2 :

Test cases 1 and 3 will get satisfied.Therefore it's not sufficient.

Combining both we can ascertain that one is even and another is odd therefore the sum will be odd.

Hence option C

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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


(1) The product of the greatest common factor of x and y and the least common multiple of x and y is even
\( x=2\) and \(y =4 \) HCF = 2 and LCM = 4 YES to the stem .
\(x=2 \)and \(y = 3 \) HCF = 1 and LCM = 6 NO to the stem.

INSUFF.

(2)The greatest common factor of x and y is 1
\(x=2 \)and \(y = 3 \) = No to the stem
\( x=3 \)and \(y = 5 \) + yes to the stem.

INSUFF.

1+2
Since HCF is odd and LCM is even we need a 2 .

Hence (2,3) (2, 5) (2, 7) (2,11) etc

Hence \(x+y\) is an odd integer.

SUFF.

Ans- C
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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

We Know

X * Y = LCM * GCF
Product of the greatest common factor of x and y and the least common multiple of x and y is even
This means
X*Y= Even

There can be both cases from this
For Example

Even*odd= Even
Even+Odd= Odd

Odd*Even= Even
Odd+Even= Odd

Even*Even= Even
Even+Even= Even

Hence this statement is not sufficient


(2) The greatest common factor of x and y is 1
If Greatest Common Factor is 1
the both cannot be even
Like
Either both will be odd or one will be even and other will be odd

Odd+Odd= Even
Odd+Even= Odd

Thus Two Cases are still possible
Not sufficient

Combining 1 and 2

It means One number is even and the other is odd
Thus We get Single case

Odd+Even= Odd

Both Statement are sufficient

Answer C

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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

GCD(x,y) * LCM (x,y) = x * Y = Even
Case 1) x=1,y=2; GCD=1 and LCM=2, GCD*LCM=2=even; x+y=1+2= 3 =odd NO
Case 2) x=2,y=2; GCD=2 and LCM=2, GCD*LCM=4=even; x+y=2+2= 4 =odd YES

This statement is insufficient

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

GCD(x,y)=1
Case 1) x=2,y=3; GCD=1, x+y=5=odd NO
Case 2) x=3,y=5; GCD=1, x+y=8=odd YES

This statement is insufficient

Both statement together:
GCD*LCM=even; 1*LCM= even; LCM=even, either x or y is having even and other number is having odd number
x+y=even+odd=odd
Answer is C

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For x+y to be even, either both x and y should be even or odd. If one is odd and the other is even the sum will be odd.

Statement 1: product of HCF of x,y and LCM of x,y is even. As product of HCF and LCM is equal to product of the numbers, x×y= even.
A product can be even when either one of the numbers if odd and the other even or both numbers are even. Hence this statement is insufficient in knowing x+y is even or odd as both cases are possible

Statement 2: HCF is 1. Again insufficient.

Considering both statement, as HCF is odd, LCM should be even for their product to be even. Hence x,y are odd and an even pair. This is sufficient. Hence answer is C.

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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

x+y can be:
Even if x and y are BOTH even or odd
If either one of x or y is odd the sum x+y will be odd

Statement 1:
HCF (x,y) * LCM (x,y) = 2m
This means that both (x,y) cannot be odd.
Possible cases:
I) one of (x,y) is even and other is odd (sum is odd)
ii) both (x,y) are even (sum is even)
Insufficient


Statement 2:
HCF(x,y) = 1
This means both (x,y) cannot be even (otherwise HCF would be 2)
Possible cases:
I) one of (x,y) is even and other is odd (sum is odd)
ii) both (x,y) are odd (sum is even)
Insufficient

Combining 1 and 2, one of (x,y) is even and the other is odd, hence sum is odd
Sufficient

Answer is C

Given: x,y>0

Question: is x+y even?

The expression x+y will be even only when both x and y are even or whe both are odd.

Statement 1: It just says that xy is even. So either x is even and y is odd;

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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

IMO C
(1) the product of GCF and LCM equal the product of x and y.
+ lets call x = a^a1 x b^b1 x c^c1.. and y = a^a2 x b^b2 x c^c2... where a,b and c are prime numbers and a1>=a2, b1>= b2 and c1<=c2.
* GCF = a^a2 x b^b2 x c^c1
* LCM = a^a1 x b^b2 x c^c2
+ GCM X LCM = a^a1 x a^a2 × b^b1 x b^b2... = xy

Xy is even so either
+ x even and y even => x + y even
+ x even and y old => x + y odd
Not sufficient

(2) GCF = 1
+ x =2 and y=3 => x+y odd
+ x=3 and y=5 => x+y even
Not sufficient

(1)(2)
Xy even & gcf =1 => x = even and y odd => x+y odd => sufficient

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The product of the greatest common factor of x and y and the least common multiple of x and y is even
Note that HCF * LCM = x*y that means for x+y to be even both xy need to be even or either of x or y could be even.
Try 2 and 1 for No . Try 2 and 2 for a Yes. Insufficient

The greatest common factor of x and y is 1
Means numbers are co prime

Try 2 and 1 for No . Try 3 and 1 for a Yes. Insufficient.

On combining, GCF is 1 means that x and y both can't be even otherwise we will get at least one 2 from each x and y , but that is not the case. Since 1 states that the LCM is even, that means X or Y , either of the values is even and the other is odd . Hence, sum of even integer and an odd integer is always ODD, so the answer to our question is No. Hence, Sufficient

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We know; odd + odd = even and even + even = even; odd*even= even and even*even = even

Also, LCM*GCD = product of two numbers (x and y in this case)

Now, coming to questions; x+y = even if x and y is even or x and y are odd

statement 1, tells us x*y = LCM*GCD = even; which means either x and y are both even or either x or y is odd the other is even. Thus statement 1 is insufficient.

statement 2, tells us GCD(x,y)=1 which is odd. but we don't know anything about x and y. so Thus statement 2 is insufficient.

Adding both statement 1 and 2;

We know x*y = LCM*1 = even*odd = even. Thus, either x or y must be even and the other one must be odd. Hence x+y= odd which is sufficient
Thus answer is C.
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Explanation:

Given: x,y are positive integer
Ti find: is x+y is even integer?

Statement 1: The product of the greatest common factor of x and y and the least common multiple of x and y is even.

Case 1: if x,y is 2,3
G.C.F of 2,3 is 1.
L.C.M of 2,3 is 6.
So, Product of G.C.F & L.C.M is 6 which is even.
So, x+y = 5 (which is odd)

Case 2: if x,y is 2,8
G.C.F of 2,8 is 2.
L.C.M of 2,8 is 8.
So, Product of G.C.F & L.C.M is 16 which is even.
So, x+y = 10 (which is even)

Not Sufficient

Statement 2: The greatest common factor of x and y is 1.

Case 1: if x,y is 2,3
G.C.F of 2,3 is 1.
So, x+y = 5 (which is odd)

Case 2: if x,y is 3,5
G.C.F of 3,5 is 1.
So, x+y = 8 (which is even)

Not Sufficient

From statement 1 & 2:

from statement 2 We know x,y cannot be both even as G.C.F is 1. (Either both odds or 1 even and another odd)
from statement 1, as we know product of G.C.F & L.C.M has to be even, then for LCM to be even we need one digit even.

Which means out x & y one has to be even and other have to be odd.

So, X+Y = odd

Sufficient

IMO-C

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