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09 Dec 2019, 01:26
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B
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Difficulty:
55%
(hard)
Question Stats:
63% (02:19) correct
37%
(02:31)
wrong
based on 132
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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
Are You Up For the Challenge: 700 Level Questions
(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
09 Dec 2019, 01:46
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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
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
Updated on: 07 Sep 2020, 10:32
Originally posted by redahansali on 12 Jul 2020, 22:02.
Last edited by redahansali on 07 Sep 2020, 10:32, edited 1 time in total.
Last edited by redahansali on 07 Sep 2020, 10:32, edited 1 time in total.
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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 evenThat 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 is inconclusive on x : x can be even or odd (take 2 and 3 as examples).
z in also unknown.
Statement 1 is insufficient.
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 sufficient.
Answer is B.
