number game again

[vaivish]

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

[pepeprepa]

That's a nice question.


If x and y are positive integers, what is the remainder when 3^(4+4x)+9^y is divided by 10?

(1) x=25
3^104 + 9^y= 9^52 + 9^y
9^52 is an integer with a unit digit which is 1. If y=1, it will give a multiple of 10 but if y=2, it will not give a multiple of 10. We have example/counter-example, it's insufficient.

(2) y=1
3^(4+4x)+9 = 9^(2+2x) + 9
Let's look at the unit digit of 9^(2+2x)
You know that 9^x has a unit digit of 1 when x is even and has a unit digit of 9 when x is odd. That's the thing to see. Given 2+2x will give you an even integer, you can state that 9^(2+2x) has a unit digit of 1. Unit digit of 1 + the 9 will give you a multiple of 10 and so the remainder is 0.
Sufficient