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 4Case 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 4Since we cannot answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: b = c² – 1No 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 4Case 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 4Since 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
4So, the answer to the target question is
YES, a² – b² IS divisible by 4Since we can answer the
target question with certainty, the combined statements are SUFFICIENT
Answer: C
Cheers,
Brent
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