If x is any positive integer, is x divisible by 2?
1) x3(x cubed) + x is divisible by 4.
2) 5x + 4 is divisible by 6.
divisibility
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I would go with D)
Stmt II
5x+4/6 = k (some integer)
x = 6k-4/5
6k-4 is always even so even/odd is even or decimal but since x is given to be a positive integer x is always divisible by 2 since its even
SUFF
Stmt I
x^3+x is divisible by 4
This means x^3 is divisible by 4 and x is diviisble by 4
So x is defnitely divisible by 2
SUFF
Stmt II
5x+4/6 = k (some integer)
x = 6k-4/5
6k-4 is always even so even/odd is even or decimal but since x is given to be a positive integer x is always divisible by 2 since its even
SUFF
Stmt I
x^3+x is divisible by 4
This means x^3 is divisible by 4 and x is diviisble by 4
So x is defnitely divisible by 2
SUFF
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cramya can you explain where all we can apply similar rulescramya wrote:I would go with D)
Stmt II
5x+4/6 = k (some integer)
x = 6k-4/5
6k-4 is always even so even/odd is even or decimal but since x is given to be a positive integer x is always divisible by 2 since its even
SUFF
Stmt I
x^3+x is divisible by 4
This means x^3 is divisible by 4 and x is diviisble by 4
So x is defnitely divisible by 2
SUFF
x^3+x is divisible by 4
"This means x^3 is divisible by 4 and x is diviisble by 4"
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Amit,
Here it goes(hope I dint add too much confusion here):
Two cases to consider here:
Case 1: x+x^3 will be divisible by 4 if x and x^3 individually are divisible by 4. If 2 components individiaully are divisible by a number then their sum definitely is
This means x is divisible by 2
Case 2: x and x^3 are both not divisible by 4 then x+x^3 will be divisible by 4 if x and x^3 remainders sum is divisible by 4
Whats unique about x^3 if I am not mistaken is that x will give the same remainder as x^3 when x^3 is not divisible by that number. Therefore their sum will never be diviisble by 4
Eg: x=3 x^3=27 when divided by 4 Both give a remainder of 3 so the sum of the remainders 6 is not divisble by 4 so x+x^3 is not divisible by 4
The 2nd CASE is not possible since x^3+x is given to be divisible by 4 so it has to be case 1
Hope this helps!
Someone feel free to correct me if I am mistaken in any of the statements made above.
I have tried to adapt the explanation to this problem (since we are dealing wiht the same number's different powers) but please see Ian's explanation to my question here:
https://www.beatthegmat.com/is-x-16-y-8- ... 26911.html
Regards,
Cramya
Here it goes(hope I dint add too much confusion here):
Two cases to consider here:
Case 1: x+x^3 will be divisible by 4 if x and x^3 individually are divisible by 4. If 2 components individiaully are divisible by a number then their sum definitely is
This means x is divisible by 2
Case 2: x and x^3 are both not divisible by 4 then x+x^3 will be divisible by 4 if x and x^3 remainders sum is divisible by 4
Whats unique about x^3 if I am not mistaken is that x will give the same remainder as x^3 when x^3 is not divisible by that number. Therefore their sum will never be diviisble by 4
Eg: x=3 x^3=27 when divided by 4 Both give a remainder of 3 so the sum of the remainders 6 is not divisble by 4 so x+x^3 is not divisible by 4
The 2nd CASE is not possible since x^3+x is given to be divisible by 4 so it has to be case 1
Hope this helps!
Someone feel free to correct me if I am mistaken in any of the statements made above.
I have tried to adapt the explanation to this problem (since we are dealing wiht the same number's different powers) but please see Ian's explanation to my question here:
https://www.beatthegmat.com/is-x-16-y-8- ... 26911.html
Regards,
Cramya
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- logitech
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If x is any positive integer, is x divisible by 2?
1) x3(x cubed) + x is divisible by 4.
X(x^2+1)
Either X or X^2+1 is divisible with 4
or they both are divisible with 2
X=even , x^2+1 can not be even so we rule out that option
X=odd NOPE
so either X or X^2+1 needs to be divided by 4
if X is divided by 4 it is divisible by 2
X^2 needs to be ODD to be divisible with 4 because there is + 1
SO SUF!
try 4,8,12, 16, and etc...
2) 5x + 4 is divisible by 6.[/quote]
means it is divisible by 3 and 2 at the same time
well if it is divisible by 2 it is an even number
5x+4 = EVEN
5x = EVEN-4
5x = ANOTHER EVEN
x can not be ODD
X can be EVEN
so X can be divided by two
SUF
Choose D
1) x3(x cubed) + x is divisible by 4.
X(x^2+1)
Either X or X^2+1 is divisible with 4
or they both are divisible with 2
X=even , x^2+1 can not be even so we rule out that option
X=odd NOPE
so either X or X^2+1 needs to be divided by 4
if X is divided by 4 it is divisible by 2
X^2 needs to be ODD to be divisible with 4 because there is + 1
SO SUF!
try 4,8,12, 16, and etc...
2) 5x + 4 is divisible by 6.[/quote]
means it is divisible by 3 and 2 at the same time
well if it is divisible by 2 it is an even number
5x+4 = EVEN
5x = EVEN-4
5x = ANOTHER EVEN
x can not be ODD
X can be EVEN
so X can be divided by two
SUF
Choose D
LGTCH
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- ronniecoleman
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Logitech, camya
appreciate your inputs!
x * ( x^2 + 1)
now if x is multiple of 4 ... case suff
now if x is odd
odd * ( odd * odd + 1)
now this has to be the odd*odd + 1 has to be the multiple of 4
now is there any rule that
odd * odd + 1 cannot be multiple of 4
PLs help
appreciate your inputs!
x * ( x^2 + 1)
now if x is multiple of 4 ... case suff
now if x is odd
odd * ( odd * odd + 1)
now this has to be the odd*odd + 1 has to be the multiple of 4
now is there any rule that
odd * odd + 1 cannot be multiple of 4
PLs help
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odd*odd + 1 may or may not be divisible by 4ronniecoleman wrote:Logitech, camya
appreciate your inputs!
x * ( x^2 + 1)
now if x is multiple of 4 ... case suff
now if x is odd
odd * ( odd * odd + 1)
now this has to be the odd*odd + 1 has to be the multiple of 4
now is there any rule that
odd * odd + 1 cannot be multiple of 4
PLs help
but here we are considering the square of a number
lets take a odd number x = 2b+1
where b is any non-negetive integer
x^2 + 1 = (2b+1)^2 + 1
(4b^2 + 4b + 1) + 1
(4b^2 + 4b + 2)
this number will always leave a rem. = 2.
hope that clears your doubt