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

Question

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

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

Bunuel · Accepted Answer

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

Given: x=ky, for some positive integer k. Question: is y = 2?

(1) y ≠ 1. Not sufficient.

(2) x + 2 is a multiple of y --> x+2=ny, for some positive integer n --> ky+2=ny --> (n-k)y=2 --> as all variables represent positive integers then either n-k=2 and y=1 (for example: x=4 and y=1) or n-k=1 and y=2 (for example: x=2 and y=2). Not sufficient.

Or: as y is factor of both x and x+2 (x/y=integer and (x+2)/y=x/y+2/y=integer) then y must be a factor of 2, so it can only be 1 or 2.

(1)+(2) As from (1) y ≠ 1 then from (2) y=2. Sufficient.

Answer: C.

subhashghosh · Answer

Hi Bunuel

One question, do you mean :

Or: as  y  is factor of both x and x+2 (x/y=integer and (x+2)/y=x/y+2/y=integer) then  y  must be a factor of 2, so it can only be 1 or 2.

instead of :

Or: as  x  is factor of both x and x+2 (x/y=integer and (x+2)/y=x/y+2/y=integer) then  x  must be a factor of 2, so it can only be 1 or 2.

Regards,
Subhash

jamifahad · Answer

Given: x mod y = 0

To prove: y =2

Stmt 1: y ≠ 1. y can take any value for different x for which x mod y = 0 will be satisfied. INSUFFICIENT.

Stmt 2: (x+2) mod y = 0. 
 x mod y + 2 mod y = 0
 0 + 2 mod y = 0 (Given x mod y = 0)
 2 mod y = 0
That means y can be either 1 or 2 for which 2 mod y = 0 will be satisfied. INSUFFICIENT.

Stmt 1 and Stmt 2 combined: 
From Stmt 1, y ≠ 1
From Stmt 2, y = 1 or y = 2

Hence together, y = 2.

Ans: C.

DeeptiM · Answer

Please help me understand how can 1 and 2 satisfy the equation 2 mod y = 0

Pls help me with the workings. Thanks.

jamifahad · Answer

Hi,

When x is a multiple of y, it means that when x is divided by y, it will leave 0 as remainder. 
It can be written x mod y = 0. 
Suppose x=6 and y = 2. Hence x is divisible by y. i.e 6 mod 2 = 0 (leaves remainder 0)

Stmt1: y ≠ 1
But you can clearly see in the number i have picked that for x=6, y can be 2,3, or 6. All the values will satisfy equation x mod y = 0. i.e 6 divided by 1,2,3,and 6 will all leave 0 as remainder. Hence y can be 2 or 3 or 6. Insufficient.

Stmt2: x+2 is a multiple of y. x+2= 6+2=8 is a multiple of 2. Hence equation x mod y = 0 is satisfied.
But pick x=6 and y=1. in this case equation x mod y = 0 is satisfied and x+2 =>6+2=8 is a multiple of 1. Hence we have different values of y for different values of x. Hence one value of y cannot be determined. Insufficient.

Take both together, from stmt1, y can be 2,3, or 6. From stmt2, y can 1,2. Together, only 2 satisfy. Hence sufficient.

mvictor · Answer

1 clearly insufficient
2 x+2 = multiple of y
it means that x is a multiple of y, and 2 is a multiple of y (N.B. any multiples of Y added gives another multiple of Y, and since x+2 is a multiple of Y, we know for sure that both of the numbers are multiple of Y)
it can be the case that y = 1 or y = 2. this statement alone is insufficient.
taken together, y is not 1, thus, y should be 2.

bumpbot · Answer

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

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