vaivish wrote:. What is the remainder when (n-1)*(n+1) is divided by 24?
(1) When n is divided by 3, the remainder is 1.
(2) n is odd.
A. Statement (1) ALONE is sufficient but Statement (2) ALONE is not sufficient.
B. Statement (2) ALONE is sufficient but Statement (1) ALONE is not sufficient.
C. BOTH Statements TOGETHER are sufficient, but NEITHER Statement alone is sufficient.
D. Each Statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.
Let's start by rewriting the question!
What is the remainder when (n^2 - 1) is divided by 24?
(1) n/3 has a remainder of 1.
If n = 4, then n^2 - 1 = 15. 15/24 = 0rem15
If n = 7, then n^2 - 1 = 48. 48/24 = 2rem0
insufficient!
(2) n is odd
if n = 3, then n^2 - 1 = 8. 8/24 = 0rem8
we already know that n=7 gives us 2rem0
insufficient!
Together:
n=1, 7, 13, ...
so, n^2 - 1 = 0, 48, 168, ..., all of which are multiples of 24. Sufficient, choose (c).
If you want the "elegant" solution, you need to have a solid understanding of factors (which is why for many test takers, picking numbers is the best way to approach number property yes/no questions).
Back to the original question:
What is the remainder when (n-1)*(n+1) is divided by 24?
24 = 2*2*2*3
So, does (n-1)(n+1) have 2, 2, 2, and 3 among it's prime factors?
(1) n/3 has a remainder of 1.
So, (n-1) is a multiple of 3. (n+1)/3 has a remainder of 2.
All we know for sure is that the expression has a factor of 3, we don't know about the 2s: insufficient.
(2) n is odd.
If n is odd, then n-1 and n+1 are consecutive even integers. All pairs of consecutive even integers have at least 3 factors of 2 between them. (Remember, "0" has an infinite number of factors.)
So, we know that we have our 3 "2"s, but we don't know if we have any "3"s: insufficient.
Together, we know that we have at least 1 factor of 3 and at least 3 factors of 2: sufficient, choose (c).
