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Re: If a, b, and c are integers and abc ≠ 0, is a2 – b2 a multiple of 4?
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11 Dec 2014, 22:03
Question can be written as a^2 - b^2 or (a-b)*(a+b) is a multiple of 4 STAT1 alone is not sufficient as we don't know anything about b STAT2 alone is not sufficient as we don't know anything about a Taking both together
If c is odd then c-1 will be even and c*c-1 will be a multiple of 2 else if c i even then again c*c-1) will be a multiple of 2
So, in both the cases c*(c-1) is a multiple of 2. So, -2*(c^2)*((c-1)^2) will be a multiple of 4 (Actually will be a multiple of 8 too) So, Answer will be C
Hope it helps!
viktorija wrote:
If a, b, and c are integers and abc ≠ 0, is a^2 – b^2 a multiple of 4?
If a, b, and c are integers and abc ≠ 0, is a^2 – b^2 a multiple of 4?
(1) a = (c – 1)^2 --> a=c^2-2c+1.
(2) b = c^2 – 1.
How about this approach? If I simplify question stem to best possible, what am I asked? Is a / b even?
Why? a. A multiple pf 4 is always even b. Squaring of odd / even is always odd /even c. Even - Even = Even. or Odd - Odd = Even
I did above in my head. Essentially, either of statements is clearly insuff since I need to know both a and b.
What happens when I combine and simplify \(a^2\) - \(b^2\) I get -2c + 1.
Even if I do not know value of c, I know -2 will always be even Even + Odd = Odd. So, we do get an unique ans: NO
Hello
Its NOT necessary for a & b to be even for this to be true. a & b could be both odd also. Eg, a=5, b=3, here a^2 - b^2 = 25-9 = 16, which is a multiple of 4.
If a, b, and c are integers and abc ≠ 0, is a^2 – b^2
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22 Apr 2018, 09:54
The math on the previous posts is clear, so I can't add anything to that. That's a surefire way of getting the problem correct.
I took a different approach in plugging in numbers to the prompts. It was obvious right away that 1 and 2 alone are insufficient because they don't give full details about the variables.
After combining, start picking smart numbers for C and work out what A and B would be. I made C = 2, 3, and 4, and then worked out what that would make A and B equal to. This gave me a good enough sample size to test divisibility of 4 and to feel comfortable enough in picking answer choice C.
Is this a perfect approach? No, but I got to the right answer in well under 2 minutes with a high probability of being correct.
Re: If a, b, and c are integers and abc ≠ 0, is a^2 – b^2
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22 Feb 2019, 09:01
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mun23 wrote:
If a, b, and c are integers and abc ≠ 0, is a² – b² a multiple of 4?
(1) a = (c – 1)² (2) b = c² – 1
Given: a, b, and c are integers and abc ≠ 0
Target question:Is a² – b² a multiple of 4?
Statement 1: a = (c – 1)² No information about b. NOT SUFFICIENT
If you're not convinced, let's TEST some values. There are several values of a, b and c that satisfy statement 1. Here are two: Case a: a = 4, b = 2 and c = 3. In this case, a² – b² = 4² – 2² = 12. So, the answer to the target question is YES, a² – b² IS divisible by 4 Case b: a = 4, b = 3 and c = 3. In this case, a² – b² = 4² – 3² = 7. So, the answer to the target question is NO, a² – b² is NOT divisible by 4 Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: b = c² – 1 No information about a. NOT SUFFICIENT
If you're not convinced, let's TEST some values. There are several values of a, b and c that satisfy statement 2. Here are two: Case a: a = 10, b = 8 and c = 3. In this case, a² – b² = 10² – 8² = 36. So, the answer to the target question is YES, a² – b² IS divisible by 4 Case b: a = 9, b = 8 and c = 3. In this case, a² – b² = 9² – 8² = 17. So, the answer to the target question is NO, a² – b² is NOT divisible by 4 Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined Statement 1 tells us that a = (c – 1)², so a² = (c – 1)⁴ Statement 2 tells us that b = c² – 1, so b² = (c² – 1)²
This means that: a² - b² = (c – 1)⁴ - (c² – 1)² KEY: the right side is a DIFFERENCE OF SQUARE, which means we can factor it We get: a² - b² = [(c – 1)² + (c² – 1)][(c – 1)² - (c² – 1)] Expand and simplify to get: a² - b² = [(c² - 2x + 1) + (c² – 1)][(c² - 2x + 1) - (c² – 1)] Simplify to get: a² - b² = [2c² - 2x][-2x + 2] Factor out some 2's to get: a² - b² = [2(c² - x)][2(-x + 1)] Simplify to get: a² - b² = 4(c² - x)(-x + 1) We can see that a² - b² is clearly divisible by 4 So, the answer to the target question is YES, a² – b² IS divisible by 4
Since we can answer the target question with certainty, the combined statements are SUFFICIENT
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73% (02:05) correct 
