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# If z and x are integers is (z^3 + x)(z^4 + x) even?

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If z and x are integers is (z^3 + x)(z^4 + x) even?  [#permalink]

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09 Dec 2019, 00:26
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45% (medium)

Question Stats:

65% (02:22) correct 35% (02:46) wrong based on 51 sessions

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If z and x are integers is $$(z^3 + x)(z^4 + x)$$ even?

(1) $$x^3 + x^2$$ is even

(2) $$\frac{z^2 + x^2}{2}$$ is odd

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Re: If z and x are integers is (z^3 + x)(z^4 + x) even?  [#permalink]

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09 Dec 2019, 00:46
determine
is $$(z^3 + x)(z^4 + x)$$ even
possible when both sum is odd / even
#1
$$x^3 + x^2$$ is even
value of z is not know ; insufficient
#2
$$\frac{z^2 + x^2}{2}$$ is odd
z^2 + x^2 = must be even i.e both are either odd or even
so
$$(z^3 + x)(z^4 + x)$$ even ; yes
sufficient
IMO B

Bunuel wrote:
If z and x are integers is $$(z^3 + x)(z^4 + x)$$ even?

(1) $$x^3 + x^2$$ is even

(2) $$\frac{z^2 + x^2}{2}$$ is odd

Are You Up For the Challenge: 700 Level Questions
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Joined: 12 Jul 2020
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Re: If z and x are integers is (z^3 + x)(z^4 + x) even?  [#permalink]

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12 Jul 2020, 21:02
I Believe Archit31110 is wrong.

Question states

If z and x are integers is $$(z^3+x)(z^4+x)$$ even ?

Analysing the question stem

The expression is even if and only if at least one of its multipliers is even
That is $$(z^3+x)$$ is even (case I) or $$(z^4+x)$$ is even (case II).

Let's look at case I :
It is even if z^3 is even and x is even, OR if z^3 is odd and x is odd.

Let's look at case II :
It is even when x is even (as z^4 will always be even no matter what the z is)

So we can make a table with x/z conditions that will give us an answer to the question stem:
Z......../......X......./......(z^3+x)(z^4+x)
even.../....odd.....=>.........odd
even.../....even....=>.........even
odd.../....even.....=>.........even
odd.../....odd.....=>.........even

So, looking at the statements, we should be targetting the parity of X and Z. Note that if we can determine that X is even, (z^3+x)(z^4+x) is automatically going to be even.

Looking at statements

Statement 1 :
x^3 + x^2 is even = x^3 is even (because x^2 is always going to be even).
This leads to x is even, which leads to (z^3+x)(z^4+x) being even.
Statement 1 is sufficient.

Statement 2 :
$$\frac{(x^2+z^2)}{2}$$ is odd => $$(x^2+z^2)$$ is even (and ends with 0, 2 or 6, so that when divided by 2, the result is odd).
Since $$(x^2+z^2)$$ is even, then both x and z either have to be both even or both odd in order to have an even sum of their respective squares.
Remember that the question stem gives us a "no" (i.e. the original expression is odd) only when x is odd and z even.
With statement 2, that is not possible, as it would lead to x^2 + z^2 = odd.

Therefore, statement 2 is also sufficient.

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Re: If z and x are integers is (z^3 + x)(z^4 + x) even?  [#permalink]

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12 Jul 2020, 21:15
Bunuel wrote:
If z and x are integers is $$(z^3 + x)(z^4 + x)$$ even?

(1) $$x^3 + x^2$$ is even

(2) $$\frac{z^2 + x^2}{2}$$ is odd

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1) x^2 (x +1) is even, x can be both even or odd. Not sufficient

2) z^2 + x^2 = 2* some integer, so z + x are even. So (z^3 +x) (z^4 + x) is even. suffcient.

Re: If z and x are integers is (z^3 + x)(z^4 + x) even?   [#permalink] 12 Jul 2020, 21:15