\(\frac{10^x+y}{y}=\frac{10^x}{y}+1\) --> so we basically need the remainder upon division 10^x by y (as in 10^x+y the second term y is divisible by y itself then only the first term matters for remainder).

(1) x=5 --> need the value of y. Not sufficient.
(2) y=2 --> as x is a positive integer then 10^x is an even number and 10^x=even divided by 2 yields the remainder of zero. Sufficient.

Answer: B.
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REM[10^(x+y)]/y

(1) x=5

REM[10^(5+y)]/y no information about y so REM is not consistent hence INSUFFICIENT.

(2) y=2

REM[10^(x+2)]/2

Any power of 10 is even => Divisible by 2 => REM =0

Hence SUFFICIENT

Hence (B) !

Given: x and y are positive integers

Target question: What is the remainder when \(10^x + y\) is divided by y?

Statement 1: x = 5
Since we have no information about y, we cannot answer the target question with certainty
Statement 1 is NOT SUFFICIENT

Statement 2: y = 2
Important: When we divide an integer by 2, the remainder will be EITHER 1 OR 0, depending on whether the integer is odd or even.
If the number is ODD, then we get a remainder of 1 after dividing by 2
If the number is EVEN, then we get a remainder of 0 after dividing by 2

Now that we know the value of y, our target question becomes, "What is the remainder when \(10^x + 2\) is divided by 2?"
Since \(10^x\) is EVEN for all positive integer values of x, we know that \(10^x + 2\) must be EVEN

So, our target question becomes, "What is the remainder when SOME EVEN NUMBER is divided by 2?"
The answer to this new target question is the remainder will be 0
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: B

Cheers,
Brent

RELATED VIDEO

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If x and y are positive intergers, what is the remainder when 10^x+y is divided by y?

Prior to looking at the items, I would first separate the 10^x and 10 ^y and divide them by y as shown below.

(10^x)(10^y)/y

This will tell us that if y is divisible by either item we would know that y is a divisor with no remainder.

(1) x=5
Tells us nothing about the divisor y.

(2) y=2

10^2 is even and divisible by 2. sufficient!