enigma123 wrote:
If n is a positive integer and r is the remainder when n^2 - 1 is divided by 8, what is the value of r?
(1) n is odd
(2) n is not divisible by 8
Given: r is the remainder when (n² - 1) is divided by 8 Target question: What is the value of r? Statement 1: n is oddLet's test some ODD values of n
If n = 1, then n² - 1 = 1² - 1 = 0, and 0 divided by 8 leaves remainder 0. So, the answer to the target question is
r = 0If n = 3, then n² - 1 = 3² - 1 = 8, and 8 divided by 8 leaves remainder 0. So, the answer to the target question is
r = 0If n = 5, then n² - 1 = 5² - 1 = 24, and 24 divided by 8 leaves remainder 0. So, the answer to the target question is
r = 0If n = 7, then n² - 1 = 7² - 1 = 48, and 0 divided by 8 leaves remainder 0. So, the answer to the target question is
r = 0 At this point, we might conclude that
r will ALWAYS be 0So, statement 1 is SUFFICIENT
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If you're not convinced, here's an algebraic solution as well:
If n is ODD, then n =
2k + 1 (for some integer value of k)
So, n² - 1 = (
2k + 1)² - 1 = 4k² + 4k + 1 - 1 = 4k² + 4k = 4(k² + k)
Notice that, if k is odd, then k² + k is EVEN, which means k² + k =
2 times some integerSo, n² - 1 = 4(k² + k) = 4(
2 times some integer) =
8 times some integerIn other words, n² - 1 is a multiple of 8, which means the answer to the target question is
r = 0Similarly, if k is even, then k² + k is EVEN, which means k² + k =
2 times some integerSo, n² - 1 = 4(k² + k) = 4(
2 times some integer) =
8 times some integerIn other words, n² - 1 is a multiple of 8, which means the answer to the target question is
r = 0In both cases, the answer to the target question is
r = 0Since we can answer the
target question with certainty, statement 1 is SUFFICIENT
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Statement 2: n is not divisible by 8 There are several values of n that satisfy statement 2. Here are two:
Case a: n = 3. In this case, n² - 1 = 3² - 1 = 8, and 8 divided by 8 leaves remainder 0. So, the answer to the target question is
r = 0Case b: n = 4. In this case, n² - 1 = 4² - 1 = 15, and 15 divided by 8 leaves remainder 7. So, the answer to the target question is
r = 7Since we cannot answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Answer: A
Cheers,
Brent
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