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Number Theory, Divisibility Q

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Number Theory, Divisibility Q

by sal_xcool » Sun Mar 13, 2011 10:31 am
If x is a positive integer, is x divisible by 2?

(1) x^3 + x is divisible by 4.

(2) 5 x + 4 is divisible by 6.

OA D


I understand why choice B alone is sufficient for this question. That's easy.
However, choice A is very tricky to me.
I understand that in order for a number to be divisible by 4, it has to be even number that has 4 as one of its factors. But how can you tell for certain that no odd number ever makes the expression in choice A divisible by 4. Sure, you can plug in few odd numbers but you can never tell for certain that the expression is not divisible by 4 when x is odd. I am looking for more systematic and concrete approach to this problem. Thanks alot for your help.
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by Night reader » Sun Mar 13, 2011 10:54 am
you are right, in hasty solution this might be tricky/tempting to choose NS for st(1)

x {integer} >0. Is x/2 ?

st(1) x(x*x +1)/4 since both x and (x*x+1) can be divisible by 4, we should deduce the following
~ x is odd, then (odd*odd + 1) is even, and the smallest odd integer is 1 - so (1*1 +1) could not be divisible by 4. Also check 3,5 and 9 --> 3^2+1, 5^2+1, 9^2+1 none is divisible by 4 BUT they are all divisible by 2!
~ x is even, simply x is divisible by 2.

st(1) could be Not Sufficient, if it was asking divisibility of (x^3+x) by 2. Here we have (x^3+x)/4, therefore either (x*x+1) is divisible by 4 OR x is divisible by 4. We can have only x could be even. Therefore st(1) is Sufficient.

st(2) (5x+4)/6. The number is divisible by 6 when it's divisible by 3 and 2. Even if (5x+4) is divisible by 3, it still must be divisible by 2. So out of (5x+4)/2 we can count 5x/2 +4/2 both should be divisible by 2 without remainders otherwise we cannot obtain whole number after (5x+4)/6. We note that 4/2 is fine, BUT 5x/2 is not so fine. 5 is not divisible by 2, hence x must be divisible by 2. This is sufficient to answer Yes on this question.

IOM D
sal_xcool wrote:If x is a positive integer, is x divisible by 2?

(1) x^3 + x is divisible by 4.

(2) 5 x + 4 is divisible by 6.

OA D


I understand why choice B alone is sufficient for this question. That's easy.
However, choice A is very tricky to me.
I understand that in order for a number to be divisible by 4, it has to be even number that has 4 as one of its factors. But how can you tell for certain that no odd number ever makes the expression in choice A divisible by 4. Sure, you can plug in few odd numbers but you can never tell for certain that the expression is not divisible by 4 when x is odd. I am looking for more systematic and concrete approach to this problem. Thanks alot for your help.
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by Anurag@Gurome » Sun Mar 13, 2011 7:43 pm
sal_xcool wrote:If x is a positive integer, is x divisible by 2?

(1) x^3 + x is divisible by 4.

(2) 5 x + 4 is divisible by 6.

OA D


I understand why choice B alone is sufficient for this question. That's easy.
However, choice A is very tricky to me.
I understand that in order for a number to be divisible by 4, it has to be even number that has 4 as one of its factors. But how can you tell for certain that no odd number ever makes the expression in choice A divisible by 4. Sure, you can plug in few odd numbers but you can never tell for certain that the expression is not divisible by 4 when x is odd. I am looking for more systematic and concrete approach to this problem. Thanks alot for your help.

Solution:
We need to know whether x is even or not.
Let us first consider statement (1) alone.
It says x^3 + x is divisible by 4.
Now, x^3 + x = x(x^2 + 1).
Now, x is either even or odd.
If x is odd, it can be written as 2n+1 where n >= 0.
So x(x^2 + 1) = (2n+1){(2n+1)^2 + 1} = (2n+1){4n^2 + 4n + 2} = 2(2n+1)(2n^2 + 2n + 1)
= 2*odd*odd which can never be divisible by 4.
Or we can say that x cannot be odd.
Or x has to be even.
So, (1) alone is sufficient.
Next, consider (2) alone.
It says that 5x+4 is divisible by 6.
Or 5x+4 is even.
Or 5x is even.
Since 5 is odd, x has to be even.
Or (2) alone is also sufficient.

The correct answer is (D).
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by RadiumBall » Mon Mar 14, 2011 1:59 am
Wow...fantastic problem and a fantastic solution...
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by sanju09 » Mon Mar 14, 2011 4:38 am
sal_xcool wrote:If x is a positive integer, is x divisible by 2?

(1) x^3 + x is divisible by 4.

(2) 5 x + 4 is divisible by 6.

OA D


I understand why choice B alone is sufficient for this question. That's easy.
However, choice A is very tricky to me.
I understand that in order for a number to be divisible by 4, it has to be even number that has 4 as one of its factors. But how can you tell for certain that no odd number ever makes the expression in choice A divisible by 4. Sure, you can plug in few odd numbers but you can never tell for certain that the expression is not divisible by 4 when x is odd. I am looking for more systematic and concrete approach to this problem. Thanks alot for your help.

(1) An odd x cannot make x^2 + 1 ever divisible by 4, because if x is an odd, then it can be written as 2 n ± 1 for some positive integer n, as x already is a positive integer. And then,

(x^2 + 1) = (2 n ± 1) ^2 + 1 = 4 n^2 ± 4 n + 2, clearly not divisible by 4.

Hence if x is a positive integer and x^3 + x = x (x^2 + 1) is divisible by 4, then x itself has to be divisible by 4 and hence by 2. Therefore sufficient

(2) If 5 x + 4 is divisible by an even number, then x must be an even or divisible by 2. Therefore sufficient


[spoiler]D[/spoiler]
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