If x and y are integers, is 3x^4 + 4y even? (1) x^3 is even (2) y^2

Question

If x and y are integers, is  3x^4 + 4y  even?

(1)  x^3  is even

(2)  y^{2x} + 3  is even

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Vij2019 · Answer

IMO, A based on following analysis:
Is  3x^4 + 4y  even? 
Which means question reduces to: Is x is even? Since, 4y will always be even.
1.  x^3  is even this implies x is even, as only even number multiple results in even irrespective of exponential power. Which means,  3x^4  is even, and hence, A is sufficient.
2.  y^2x + 3  is even ---->  y^2x   will be odd, since Odd + Odd is even. Hence, Y is odd and x can be either be odd or even. Hence, this is not sufficient.

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preetamsaha · Answer

If x and y are integers, is 3x^4+4y even?

(1)x^3 is even
so, x= even (2^3=8 =even)
so,x^4 = even 
3x^4 = even

now, if y = even, 4y = even
also, if y = odd, 4y = even

so, always 3x^4+4y even (even+even= even )  sufficient

(2) y^2x+3 is even
so, y^2x= odd 
or, case I : y= even x= even y^2x= even ( it is not the case) 
case II : y= even x= odd y^2x= even ( it is not the case) 
case III : y= odd x= even y^2x= odd ( it is the case) (1^4=1 odd)
case IV : y= odd x= odd y^2x= odd ( it is the case) (1^2=1 odd)

so, x may be even or odd 
so,x^4 may be even or odd
so,3x^4 may be even or odd
so,3x^4 + 4y may be even or odd (4y always even)  insufficient

correct answer A

Archit3110 · Answer

target  3x^4 + 4y  even?
possible when x is even
#1
 x^3  is even
would be possible only when x is even so sufficient
#2
  y^{2x} + 3  is even
value of y has to be odd ; but x can be even or odd so  3x^4 + 4y  may or may not be even
insufficient
OPTION A

CoachNikhilesh · Answer

The question easy is enough we have to find out  3x^4 + 4y  is even or not.

If both  3x^4  and  4y  will be odd or if both will even then their sum will be even as well.
Now, since  4y  is 4 multiple, it will always be even. Therefore, we can ignore y and focus only on  3x^4  or x

Now, let's take a look at (1)

x^3  is even. If an integer's cube is even then that integer has to contain 2. Therefore,  x^4 will be even as well.
Hence,  3x^4  will be even. (Even*Odd=Even)

We know,
 3x^4  is even and  4y  is always even.
Therefore,  3x^4 + 4y  is even. (Even + Even = Even)
 Sufficient

Next, let's take a look at (2)
 y^{2x} + 3  is even

As previously discussed , it doesn't matter what y is.
Therefore,  insufficient

stne · Answer

Dear Moderator,

Can we have the OE for this one if the OA is C, every one seems to be getting A as the answer. Thank you.

Bunuel · Answer

___________________________
The OA is A. Edited. Thank you.

NitishJain · Answer

3x^4+4y even?

4y will always be even as even* odd/even = even
This will be even only when 3x^4 is even (Even + Even = Even). 3x^4 will be even only when x is even as odd(3) * Even= Even.

This question generally asks: is x Even?

A) x^3 is even: This will only happen if x is Even. If x be odd then odd^odd = odd hence not possible. Sufficient

B)y^2x + 3 is even: This only means y^2x is odd (odd + odd = Even). This implies y is odd (Odd raise to even poser is odd). From here the nature of x cannot be known.
Insufficient.

Hence A.

bumpbot · Answer

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If x and y are integers, is 3x^4 + 4y even? (1) x^3 is even (2) y^2

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