If x and y are positive integers, is 2x a multiple of y?

Question

(1) 2x + 2 is a multiple of y
(2) y is a multiple of x

Fistail · Answer

From a:

If x = 2 and y = 3, (2x+2) is a multiple of y but not 2x. 
If x = 2 and y = 2, (2x+2) and 2x are multiple of y.

So not sufficient.

From b:  y is a multiple of x means x is equal to or smaller than y.

If y = x = 3, 2x is a multiple of y.
If y = 3 and x = 1, 2x is not a multiple of y. not sufficient.

From a and b:  (2x + 2) is a multiple of y and y is a multiple of x.

If x = 2 and y = 2, (2x+2) is multiple of y and y is also a multiple of x. So 2x is also a multiple of y.
If y = 6 and x = 2, (2x+2) is multiple of y and y is also a multiple of x but 2x is not a multiple of y.

So together also not sufficient. E.

meshell · Answer

statement 1. just means that y is even, but it does not tell us anything about the direct relationship between x and y. Simplified is really saying 2x + 2 =yk , k being an unknown multiplication of y

as the earlier post did, plug numbers.

2(2) + 2 = yk : when x is 2, y can be 2 multiplied 3 times, and so 2x will be a multiple of y BUT we could also assume y is 3 multiplied 2 times, in which case 2x ie. 4 is not a multiple of 3. this gives a yes/no answer so its insufficient.

2(x) + 2 = y (k times) 
2(2) + 2 = 2 (3 times) ...........4 is a multiple of 2
BUT
2(2) + 2 = 3 (2 times)...........4 is  not  a multiple of 3

bodleyev · Answer

x, y > 0 , integers

"Is  \frac{2x}{y}  an integer?"
"Is  2x  a multiple of (divisible by)  y ?"
"Is  y  a factor of  2x ?"

Statement 1) This statements says that  \frac{2x + 2}{y} = \frac{2(x+1)}{y}  is an integer. In other words,  2(x+1)  is a multiple of  y .

In order for both  2x  and  2(x+1)  to be divisible by  y ,  y  must be either  1  (factor of  2  and both  x  and  (x+1) ) or  2  (factor of  2 ). Thus if  y  equals  1  or  2 , then  \frac{2x}{y}  is an integer. If  y  has any other value under this constraint,  \frac{2x}{y}  is not an integer.

You can also break this statement into two expressions:  \frac{2x + 2}{y} = \frac{2x}{y} + \frac{2}{y} . If  \frac{2x}{y} + \frac{2}{y}=integer , then in order for  \frac{2x}{y}  to be an integer,  \frac{2}{y}  must also be an integer. Thus if  y  equals  1  or  2 , then  \frac{2x}{y}  is an integer. If  y  has any other value,  \frac{2x}{y}  is not an integer.

Insufficient.

Statement 2) This statement says that  \frac{y}{x}  is an integer. So  y = x*k , where  k  is some integer.

In order for  \frac{2x}{y} = \frac{2*x}{k*x}  to be an integer,  \frac{2}{k}  must be an integer, ie  k  must equal either  1  or  2 . Thus if  y = 2x  or  y = x ,  2x  will be a multiple of  y . With any other value of  k ,  2x  will not be a multiple of  y .

Insufficient.

Combined)
From statement 1,  \frac{2x}{y}  is an integer when  y={1,2} , and not an integer when  y  has another value.
From statement 2:  \frac{2x}{y}  is an integer when  y=2x  or  y=x , and not an integer when  y  is a different multiple of  x  than  2 .

So under the constraints of both statements,  \frac{2x}{y}  is an integer when  (x,y)  =  (1,1) ,  (2,2) , or  (1,2) .

It only takes one example that follows the two constraints but is not one of these 3 possibilities to prove the combined statements insufficient. A couple examples are  (x,y) = (1,4) , or  (2,6) .

E

kp1811 · Answer

stmnt1:

If y= 1 x=2 then 2x+2 = 6 and multiple of y. 2x =4 and multiple of y
but if y=3 and x=2 then 2x+2=6 and multiple of y but 2x=4 and not a multiple of y. Hence insuff

stmnt 2:

y = 2 and x= 1 then 2x is multiple of y
y= 3 and x= 1 then 2x is not multiple of y. Hence insuff

taking together
x=1 and y =2 we have y is multiple of x, 2x+2 = 4 multiple of y and 2x= 2 is multiple of y (2) . 
x=1 and y =4 we have y is multiple of x, 2x+2 = 4 multiple of y but 2x= 2 is not multiple of y (4)

Hence E

honchos · Answer

Bunuel,

w/o plugin   is their a better way to solve this question.

Bunuel · Answer

It's easy to get a NO and an YES answers with number plunging, so yes, this way is probably the fastest/easiest.

KarishmaB · Answer

Use some logic and some number plugging:

Question: Is 2x a multiple of y?

(1) 2x+2 is a multiple of y

(2x + 2) is a multiple of y. 
2x and (2x+2) are two consecutive even numbers and hence share only 2 factors: 1 and 2
So 2x will also be a multiple of y only if y is 1 or 2. Not sufficient.

(2) y is a multiple of x
Here are the multiples: x....y....2x
If y is x or 2x, 2x will be a multiple of y, else it will not be. Not sufficient.

Using both together:
If y = 2 and x = 2, all conditions are met and 2x IS a multiple of y.
If y = 6 and x = 2, all conditions are met but 2x IS NOT a multiple of y.
Not sufficient.

Answer (E)

sunaimshadmani · Answer

What if x=1 and y=2..it satisfies all the conditions??

sunaimshadmani · Answer

Statment 1 tells that y is either 1 or 2.
Statment 2 shows that y is 2 as its a multiple of x and 1 cant be a multiple of any number, thats what understand, tell me where am going wrong!

Bunuel · Answer

From (1), from 2x+2 is a multiple of y, we don't have that y is either 1 or 2. It can take many other values.

Is 2x a multiple of y?

(1) 2x+2 is a multiple of y. If x=1 and y=1, then the answer is YES but if x=1 and y=4, then the answer is NO. Not sufficient.

(2) y is a multiple of x. If x=1 and y=1, then then the answer is YES but if x=1 and y=4, then the answer is NO. Not sufficient.

(1)+(2) The same here: if x=1 and y=1, then then the answer is YES but if x=1 and y=4, then the answer is NO. Not sufficient.

Answer: E.

bumpbot · Answer

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If x and y are positive integers, is 2x a multiple of y?

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