Data Sufficiency

Hello,

I could use some help. I've been working on number properties today in the guidebook I purchased specifically for NPs. I admit that NPs are a very large weakness for me, so the errors are probably on met, but I've gone over the following three problems several times using DS strategies and my answers don't match the guidebook. I went to the errata website and no errors were listed for the odd/even CHPT. Finding three unlisted errors in one CHPT has got to be me, but I'd like to see what the rest of the forum comes up with their own answers to see if it matches mine or the book. Thanks for your help.

1. If x>1, what is the value of x?

1) There are x unique factors of x.
2) The sum of x and any prime number larger than x is odd.

2. What is the remainder when a is divided by 4?

1) a is the square of an odd integer.
2) a is a multiple of 3.

3. This is not a DS question, instead you were to answer if x is odd, even, or can't be determined & ALL VARIABLES ARE INTEGERS -
- If x/y yields an odd integer, what is x?

Q1. x > 1 (given)
what is x?

ST1: there are x unique factors of x
lets try a few values for x
if x = 2 (as x>1), factors = 1,2 (satisfies)
if x=3, factors= 1,3 (DOES NOT SATISFY as there are only 2 unique factors of X and X = 2)
x = 4, factors= 1,2,4 (DOES NOT SATISFY as there are 3 unique factors of X and x = 4)
There is no numbers besides 2 that will satisfy this.

hence St1 is sufficient.

St2: sum of x and any prime larger than x = odd
Odd = odd + Even or Odd = Even + odd
if x = 2, prime no = 3 sum = odd
but if x = 3, prime no = 5 (odd) then sum = even (odd + odd = even)
not sufficient

Hence IMO it should be A

Q2. remainder when a / 4 ?

ST1: a = sq of an odd integer
if a = 1, a^2 = 1 and 1/4 = rem 1
if a = 3, a^2 = 9 and 9/4 = rem 1
if a = 5, a^2 = 25 and 25/4 = rem 1

Sufficient

ST2: a = multiple of 3
if a = 3, 3/4 = rem 3
if a = 6, 6/4 = rem 2
Not Sufficient

Hence Ans A

Q3. if x/y = odd integer, it could either be 9/3 = 3 (where x = odd) or 6/2 = 3 (where x = even)
only with the above information, with no information about y, we cannot for certain say anything about x