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good one

by vaivish » Tue Aug 26, 2008 8:03 am
If x and y are positive integers, what is the remainder when 3^(4+4x)+9^y is divided by 10?
(1) x=25
(2) y=1
A. Statement (1) ALONE is sufficient but Statement (2) ALONE is not sufficient.
B. Statement (2) ALONE is sufficient but Statement (1) ALONE is not sufficient.
C. BOTH Statements TOGETHER are sufficient, but NEITHER Statement alone is sufficient.
D. Each Statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.


oa IS B.
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Re: good one

by sudhir3127 » Tue Aug 26, 2008 8:19 am
vaivish wrote:If x and y are positive integers, what is the remainder when 3^(4+4x)+9^y is divided by 10?
(1) x=25
(2) y=1
A. Statement (1) ALONE is sufficient but Statement (2) ALONE is not sufficient.
B. Statement (2) ALONE is sufficient but Statement (1) ALONE is not sufficient.
C. BOTH Statements TOGETHER are sufficient, but NEITHER Statement alone is sufficient.
D. Each Statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.


oa IS B.
My Answer is B.

Though my answer may not be useful to u if u arent well-versed Chinese Remainder theorem.. But for all those who understand it ..

3^(4+4x)+9^y

Statement 1. X= 25

3^104 + 9^Y/10

3^104/10 ~ 9^52/10 ~ (-1)52 = 1

9^y will depend on whether y is even or odd..

hence in sufficient

Statement 2. Y =1

3^4+4x + 9^1/10

3^4+4x/10 will give a remainder 1
9/10 = -1

+1-1=0
hence the remainder is 0.

hence sufficient.

Please let me know if u have any doubts.. thought i am sure ..if u dont know CRT its very difficult to understand my method..
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Re: good one

by anju » Tue Aug 26, 2008 8:24 am
Sudhir, can you pls. explain CRT?

Thanks
sudhir3127 wrote:
vaivish wrote:If x and y are positive integers, what is the remainder when 3^(4+4x)+9^y is divided by 10?
(1) x=25
(2) y=1
A. Statement (1) ALONE is sufficient but Statement (2) ALONE is not sufficient.
B. Statement (2) ALONE is sufficient but Statement (1) ALONE is not sufficient.
C. BOTH Statements TOGETHER are sufficient, but NEITHER Statement alone is sufficient.
D. Each Statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.


oa IS B.
My Answer is B.

Though my answer may not be useful to u if u arent well-versed Chinese Remainder theorem.. But for all those who understand it ..

3^(4+4x)+9^y

Statement 1. X= 25

3^104 + 9^Y/10

3^104/10 ~ 9^52/10 ~ (-1)52 = 1

9^y will depend on whether y is even or odd..

hence in sufficient

Statement 2. Y =1

3^4+4x + 9^1/10

3^4+4x/10 will give a remainder 1
9/10 = -1

+1-1=0
hence the remainder is 0.

hence sufficient.

Please let me know if u have any doubts.. thought i am sure ..if u dont know CRT its very difficult to understand my method..
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Solution

by mayur00 » Tue Aug 26, 2008 10:13 am
E = 3^(4+4X) + 9^Y

= (3^4 x 3^4X) +9^Y

= (9^2 x 81^X) + 9^Y

=9 ( 9 x 81^X + 9^(Y-1))

= 9 ( R + T)

where, R= 9 x 81^X

Since X = +ve integer, therefore units digit of 81^X is always 1 which means units digit of R is always 9

T = 9 ^ (Y-1)

Units digit of T can be 1...9 depending on Y.


So really the only unknown is the value of Y as the units digit itself of a number determines what the remainder is when divided by 10.


Since in (B) Y=1

Therefore,

T = 9^0 = 1

and

R = something ending with 9


There R+T will have 0 as the last digit

Therefore, E = 9x (R + T) => something with last digit 0, hence divisible by 10.


(A) doesn't really help as we need the value of Y.



I know it sounds complicated but it is really very simple. You can possibly do all of this mentally.
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by ddm » Tue Aug 26, 2008 10:44 am
can someone explain this please....
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Re: good one

by Ian Stewart » Tue Aug 26, 2008 10:48 am
vaivish wrote:If x and y are positive integers, what is the remainder when 3^(4+4x)+9^y is divided by 10?
(1) x=25
(2) y=1
We want the remainder when we divide by 10, which is simply the units digit. We can use the fact that the units digit of a product comes from the product of the units digits, and the units digit of a sum comes from the sum of the units digits:

Notice that 3^(4+4x) = (3^4)^(x+1) = 81^(x+1). No matter what x is, 81^(x+1) must end in 1 if x is a positive integer.

On the other hand, 9^y could end in 9 if y is odd, or 1 if y is even.

Thus, the units digit of 3^(4+4x)+9^y will be 0 if y is odd, and 2 if y is even. Statement 2) tells us that y is odd, so is sufficient. Statement 1 is not useful. B.
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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by malolakrupa » Tue Aug 26, 2008 1:43 pm
3^(4+4x)+9^y

Let A = 3^(4+4x) and B = 9^y . Now 3 raised to any multiplier of 4 will have 1 in its units digit .

Hence x = 25 by is not sufficient as the answer will depend on the value of y.

when y = 1 we will know for sure that the number will be divisible by 10 .

Hence choose B.
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