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16 Feb 2022, 06:30
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B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
25%
(medium)
Question Stats:
75% (01:35) correct
25%
(01:50)
wrong
based on 118
sessions
History
Date
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Not Attempted Yet
If a and b are integers and a is odd, is b even?
(1) \((a^3 + 1)*b\) is even.
(2) \(\frac{a^2}{b} = b - 4\)
(1) \((a^3 + 1)*b\) is even.
(2) \(\frac{a^2}{b} = b - 4\)
16 Feb 2022, 07:20
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Bunuel
Some important rules:
#1. ODD +/- ODD = EVEN
#2. ODD +/- EVEN = ODD
#3. EVEN +/- EVEN = EVEN
#4. (ODD)(ODD) = ODD
#5. (ODD)(EVEN) = EVEN
#6. (EVEN)(EVEN) = EVEN
Given: a and b are integers and a is odd
Target question: Is b even?
Statement 1: \((a^3 + 1)b\) is even.
If a is ODD, then a³ is ODD, which means a³ + 1 is EVEN.
So, we can rewrite statement 1 as: (EVEN)(b) = EVEN
We can see that if b is ODD, the product will be even, and if b is EVEN the product will still be even.
Since we can’t answer the target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: \(\frac{a^2}{b} = b - 4\)
Multiply both sides of the equation by b to get: a² = b(b - 4)
Expand: to get: a² = b² - 4b
Add 4b to both sides: a² + 4b = b²
If a is ODD, then a² is ODD
Since 4 is even, we know that 4b is EVEN
So our equation becomes: ODD + EVEN = b²
Since odd + even = odd, our equation becomes: ODD = b²
This means we can be certain that b is ODD, which means the answer to the target question is NO, b is definitely not even.
Since we can answer the target question with certainty, statement 2 is SUFFICIENT
Answer: B
Cheers,
Brent
17 Feb 2022, 05:40
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Bunuel
Given: a = odd
(1) Since a is odd, a^3 will also be odd.
hence, a^3 + 1 = odd + odd = Even
hence, even * b = even
b could be either even or odd.
Not Sufficient
(2) (a^2)/b = b - 4
we have a^2 = b(b-4)
since a is odd, hence, a^2 will be odd too
therefore, b(b-4) is odd too
product of 2 values is odd only when both the values are odd
and since 4 is odd,
b - 4 will be odd when b is odd
if b = even, b-4 = even and hence a^2 = b(b-4) = even
We have b = odd
the answer to the question is a definite NO, hence statement 2 is sufficient.
Answer: B
