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..

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..

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.

can someone explain this please....

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.