OFFICIAL SOLUTION:

If both x and y are positive integers, what is the remainder when \(2*10^x + 11y\) is divided by 9?

For an integer to be divisible by 9, the sum of its digits must be divisible by 9. Now, the sum of the digits of 2*10^x (20, 200, 2,000, ...) is always 2, irrespective of x. Thus we need only the value of 11y to determine divisibility of \(2*10^x + 11y\).

(1) x = 6. Not sufficient.

(2) y = 8. In this case 11y = 88, so the sum of the digits of \(2*10^x + 11y\) is 2 + (8 + 8) = 18 = {a multiple of 9}, therefore \(2*10^x + 11y\) IS divisible by 9. Sufficient.

Answer: B.
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a) 11y/9 would give different remainder for different values of y. Not sufficient.
b) 2*10^x /9 always yields 2 as a remainder. Sufficient.
Hence B

Divisibility rule of 9
"If the sum of digits of the given number is divisible by 9 then the number itself is divisible by 9."

2*10^x+11y

Stmt 1: x = 6

Whatever the value of x (x is a positive integer) sum of digits of 2*10^x is always 2. (Eg: when 20, 200, 2000 etc)
Depending on the value of y the sum of digits may or may not leave a remainder.
(For eg : if y = 4, 2*10^x+11y = 2000044, sum of digits = 10; not divisible by 9; if y = 8, 2*10^x+11y = 2000088, sum of digits = 18; divisible by 9;
As long as we don't know the value of y we cant say if the expression will leave a remainder or not.
Not suff

Stmt 2: y = 8
11y = 88 and sum of digits = 16. Whatever the value of x (x is a positive integer) sum of digits of 2*10^x is always 2.
So sum of digits of 2*10^x+11y is always 18 when y =8. Remainder will be zero.

Suff


Answer : B

Ambarish
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Hello all

My attempt:

I like to solve such questions by finding out remainders in the given equation and reducing the remainders successively.

First term
\(10^x mod (9) = 1\)
Therefore remainder of the first term will always be \(2\) irrespective of the value of \(x\)

Second term
We need to know the value of \(y\) to find out the remainder of this term \(11 mod(9) = 2\) but we need to know \(y\).

Thus statement 2 is required to answer this question. Answer I will go for is B

First of all, a C answer here seems like a trap. There must be a value where the remainder will be unchanged given one of the variables

(1) x = 6
testing values out, the remainder is not dependent on the value of x
Insufficient

(2) y = 8
No matter the value of x, the remainder is constant when y is given
Sufficient

Answer: B

( 2∗10 ^x)/9 remainder will always be 2.
Plug in value of x = 0 then remained is 2, X = 2 200/9 remainder is 2.

11y/9 can be broken as (9y +2Y)/9 hence remainder will depend on value of Y only not on X.

Hence Option B.

Ans: B
we know that 2*10^x, when divided by 9, always gives 2 as reminder. so this question requires us to find the value of y.
Solution:
1) remainder does not depend on the value of x so [Not Sufficient]
2) now from this we know the value of y so we can find the exact remainder. [Sufficient]
Statement 2 alone is sufficient to ans the question.
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This question check focus and presence of mind more than it checks mathematical concept.

Rem (10/9) = 1
Rem (100/9) = 1
Rem (1000/9) = 1

For any positive integer value of x, the remainder Rem ( 2*10^x) is same = 2

In order to have the complete value of the equation, we need y.

Statement 1 is reinstating the fact that is already deduced from the premise. Hence unnecessary and insufficient as it does not give any detail about y

Statement 2 gives a value for y

Hence B
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