voodoo_child wrote:If x is a positive integer, is x divisible by 2?
(1) x^3 + x is divisible by 4.
(2) 5x + 4 is divisible by 6.
-Voodoo
Statement 1: x^3 + x is divisible by 4
When we factor the expression we see that x(x^2 + 1) is divisible by 4
For x(x^2 + 1) to be divisible by 4, there are essentially 3 possible cases to consider:
case a: x is divisible by 4 (in which case x
is divisible by 2)
case b: x is divisible by 2, and x^2 + 1 is divisible by 2 (in which case x
is divisible by 2)
case c: x^2 + 1 is divisible by 4, and x is not divisible by 2
case b is impossible, for the following reason: If x is even (i.e., divisible by 2) then x^2 is also even, in which case x^2 + 1 is odd. If x^2 + 1 is odd, it cannot be divisible by 2
case c is impossible for the following reason: First, if x^2 + 1 is divisible by 4, then x^2 + 1 is even, in which case x^2 is odd. If x^2 is odd, then x must be odd.
Now, if x must be odd, it is not possible for x^2 + 1 to be divisible by 4
We know this because, if x is odd, we can say that x = 2k+1 for some integer k (all odd numbers can be written as 2k+1)
This means that:
x^2 + 1 = (2k+1)^2 + 1
= 4k^2 + 4k + 2
= 4(k^2 + k) +
2
Since 4(k^2 + k) is definitely divisible by 4, we can see that 4(k^2 + k) +
2 is NOT divisible by 4, which means x^2 + 1 is NOT divisible by 4.
If x^2 + 1 is NOT divisible by 4, we can eliminate case c, which means case a MUST be true, which means x MUST be divisible by 2.
So, statement 1 is sufficient.
Statement 2: 5x + 4 is divisible by 6
If 5x + 4 is divisible by 6, we know that 5x+4 must be divisible by 2, which means 5x+4 is even
If 5x+4 is even, 5x must be even (since EVEN + EVEN = EVEN)
If 5x is even, x must be even (since ODD x EVEN = EVEN)
If x is even, then x is divisible by 2
So, statement 2 is sufficient, and the answer is
D
