Data Sufficiency

242. If x and y are positive integers, is x - y divisible by 4?
(1) xy is divisible by 16.
(2) x is divisible by 4.

[spoiler]Answer: E[/spoiler]
S1)
xy can be 16 = (1,16), (2,8), (4,4)
xy can be 32=(1, 32), (2, 16), (4, 8)
xy can be 48=(1, 48), (2, 24), (3, 16), (4, 8)

(x-y) can be divisible by 4, and not divisible by 4. Thus insufficient.

S2) x is divisible by 4. Then x must be 4a. No information on y. Thus insufficient.

If x is 4, then the pairs: (1, 16) and (2,8) are possibilities. Looking at (x-y): 16-1 is not divisible by 4, but (8-4) is divisible by 4. Thus insufficient.

Is there an easier way to do this other than listing the factor pairs?


There are 20 books in a bookcase. If one book is selected at random, what is the probability that the book is either a hardback or a novel?
(1) 8 books in the bookcase are novel books and 10 books are hardbacks.
(2) 3 books in the bookcase are hardback novels.

[spoiler]Answer: C[/spoiler]
S1: We know that there are 8 novels and 10 hardbacks, but we do not know how many books are both. Insufficient.
S2: We know there are 3 books which are both hardbacks and novels. Thus 20-3=17 must be the number of books which are either hardbacks or novels. Thus 17/20 is the answer. Sufficient.

Can anyone explain way I need statement 1?

256. If n is a positive integer between 30 and 60, inclusive, what is the value of n?
(1) When n is divided by 4, the remainder is 1.
(2) When n is divided by 5, the remainder is 2.

[spoiler]Answer: E[/spoiler]

30≤n≤60

S1: n=4a+1
4(8)+1=33
4(9)+1= 37
4(10)+1=41
4(11)+1=45
4(12)+1=49
4(13)+1=53
4(14)+1=57
Insufficient
S2: n= 5b+2
5(6)+2=32
5(7)+2=37
5(8)+2=42
5(9)+2=47
5(10)+2=52
5(11)+2=57
Insufficient.

Together: 37 and 57 is in both sets of numbers. Thus insufficient.
Again, does anyone have a better method than listing out the possibilities?