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

Question

(1) y^2 + y is not a factor of x.
(2) x^3 + x is not a multiple of y.

septwibowo · Accepted Answer

Got some enlighment here

Question ask whether \frac{x}{y}=integer . Statement 1 - We can write the statement into \frac{x}{(y^2)+y}=non integer - Simplify equation, we get \frac{x}{y(y+1)}=non integer --> we isolate \frac{x}{y} --> \frac{x}{y}=non integer*(y+1) - From here, we can conclude that \frac{x}{y} can be either an integer or not an integer. - Case 1 : let say we have non integer \frac{1}{3} and y=2 , so \frac{1}{3}*(2+1) = integer . - Case 2 : let say we have non integer \frac{1}{3} and y=3 , so \frac{1}{3}*(3+1) = non integer . - INSUFFICIENT. Statement 2 - We can write the statement into \frac{x^3+x}{y}=non integer - Simplify equation, we get \frac{x(x^2+1)}{y}=non integer --> we isolate \frac{x}{y} --> \frac{x}{y}=\frac{non integer}{(x^2+1)} . - From here, we can conclude that the result of \frac{x}{y} always non integer. There is no way a non integer when divided by integer become integer . - Thus, we got the definite answer here! - SUFFICIENT B

niks18 · Answer

The re-framed question can be Is  x = yk  (where  k  is any constant)

Statement 1 : this implies that  y^2 + y = y(y+1)  does not divide  x . But  x  may or may not be a multiple of  y . 
If  x  is a multiple of  y  i.e let  x= yk , then as per the statement  \frac{x}{y(y+1)}  is not divisible, or  \frac{yk}{y(y+1)} 
or  \frac{k}{(y+1)}  or  k  is not divisible by  (y+1) . But  x  can be a multiple of  y . Similarly  x  may not be a multiple of  y  altogether. Hence the statement is not sufficient

Statement 2 : this implies  x^3+x  or  x(x^2+1)  is not a multiple of  y . This statement is Sufficient to prove that  x  is not multiple of  y  because if  x  had been a multiple of  y  then let  x=yk 
so  x(x^2+1) = yk({y^2k^2}+1) =  multiple of  y , which is not true as per the statement. Hence Sufficient

Option  B

septwibowo · Answer

Interesting question! Trying to solve twice and still not get the right answer.

Waiting for OE.

LeoN88 · Answer

Good question to clarify few pointers regarding factors and multiples.

Let's start with statement 2.
x(x^2 + 1) is not divisible by y. Sufficient to say that x alone will also not get perfectly divisible by y and hence x is Not a multiple of y.

statement 1 is confusing.

It says y(y+1) is not a factor of x, y(y+1) can't divide x perfectly.

y (y+1) x
2 3 4; 4 is not divisible by 6 but 4(x) is divisible (multiple) of 2
5 6 4; in this case 4 is also not a multiple of 6. This statement tells us nothing about whether x is a multiple of y or not.

prabsahi · Answer

Prompt:  Is x/y = Integer(k)

STAT1:

y^2 + y is not a factor of x .This implies x is not a multiple of y^2 + y

So, x/y*(y+1) is not an integer.. (Be careful of the form when distributed factors are in denominator)

Lets say x= 25 and y=5 and y+1 = 6

True case: 
Now 25/5*6  Here x/y is integer  but x/y*(y+1) is not an integer as staement 1 requires
 False Case: 
Now 25/8*9  Here x/y is not integer  and x/y*(y+1) is also not an integer as staement 1 requires

Now STAT2: 
x^3 + x is not a multiple of y.

This seems easy.

x*(x^2 +1) /y is not an integer

Now  x^2 +1  =  Integer

So for (x /y)* Integer = Non integer

means x/y definitely has to be a non integer.

Hope it helps!!

ShankSouljaBoi · Answer

Any specific approach for such questions ?  IanStewart  ?

IanStewart · Answer

I wouldn't really say that this problem is one representative example from a larger category of problem. It's somewhat typical though of many high-level GMAT problems in that it's unusual -- it's not in a standard format where you could memorize and apply a method -- but it still doesn't require any advanced knowledge to solve.

I can explain how I'd think through the problem, but I don't think you'd be able to abstract from there an 'approach' that you could apply to many other questions:

When I see Statement 1, it doesn't look sufficient -- the first thing I notice is that Statement 1 will be true any time y^2 + y is larger than x. Then x can be a multiple of y or not. I'd quickly confirm with simple numbers that Statement 1 is not sufficient: if y = 2, then y^2 + y = 6. Now x can be 4, and x is a multiple of y, or x can be 5, and x is not a multiple of y.

For Statement 2, if x is a multiple of y, then x(x^2 + 1) is always going to be a multiple of y, because it's a multiple of x. So if x(x^2 + 1) = x^3 + x is *not* a multiple of y, there's no way that x alone could be, and Statement 2 is sufficient, and the answer is B.

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

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

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