MathRevolution
What is the remainder, when n(n+2) is divided by 24 for a positive integer x?
1) n is an even integer
2) n has remainder 0 or 1 when it is divided by 3.
Asked: What is the remainder, when n(n+2) is divided by 24 for a positive integer x?
1) n is an even integer
n = 2k
n(n+2) = 2k(2k+2) = 4k(k+1)
The remainder, when n(n+2) is divided by 24 = {8,0}
NOT SUFFICIENT
2) n has remainder 0 or 1 when it is divided by 3.
n = 3k or n = 3k+1
If n = 3k ; n(n+2) = 3k(3k+2) = 9k^2 + 6k
The remainder, when n(n+2) is divided by 24 = {15,0,3}
If n=3k+1; n(n+2) = (3k+1)(3k+3) = 3(k+1)(3k+1)
The remainder, when n(n+2) is divided by 24 = {0,15,12}
NOT SUFFICIENT
(1) + (2)
1) n is an even integer
n = 2k
n(n+2) = 2k(2k+2) = 4k(k+1)
The remainder, when n(n+2) is divided by 24 = {8,0}
2) n has remainder 0 or 1 when it is divided by 3.
n = 3k or n = 3k+1
If n = 3k ; n(n+2) = 3k(3k+2) = 9k^2 + 6k
The remainder, when n(n+2) is divided by 24 = {15,0,3}
If n=3k+1; n(n+2) = (3k+1)(3k+3) = 3(k+1)(3k+1)
The remainder, when n(n+2) is divided by 24 = {0,15,12}
Combining, we get
The remainder, when n(n+2) is divided by 24 = 0
SUFFICIENT
IMO C