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

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.

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

Pls help me with the workings. Thanks.

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.

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.

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