If p and q are integers, can (q − 1) always be expressed as an integer

q can be 5 and p can be 4 --> yes
q can be 2 and p can be 6 --> no
q can be 5 and p can be 3 --> no

Answer is C

Statement 1(p>q)
Case-1 - Let p=5 and q=-4, then (q-1=-5 is (-1) times p or 5)
Case-2 - Let p=7 and q =5, then (q-1=4 cannot be expressed as a multiple of p or 7)
Hence, Statement 1 is not sufficient.

Statement 2
Case-1 - Let p=7 and q=8, then (q-1=7 is (1) times p or 7)
Case-2 - Let p=7 and q =9, then (q-1=8 cannot be expressed as a multiple of p or 7)
Hence, Statement 2 is not sufficient.

Both Statements together
Here q is positive and >1 & p is also positive and greater than 2 (since both p & q are intergers)
Thus, p is always greater than q-1, since p & q are positive and p>q.
A small number can never be expressed as an integer multiple of a large number.
Hence, both statements together are sufficient.

We have to find whether it is possible to express (q-1) as an integer multiple of p ==> (q-1) = p * k (k is some integer multiple)

Statement 1 says p>q.
Let q= 1 and p= 4
1-1 = 0 ==> 4 * 0, for k=0 --> possible
Let q=2 and p=4
2-1 = 1 But 1 cannot be expressed as a product of 4 * k, for some integer k. --> not possible
Insufficient

Statement 2 says q>1
Nothing has been given about the value of p
Let q=2 and p=1
2-1= 1 ==> can be expressed as 1*1, for k=1 --> possible
Let q=2 and p=3
2-1=1 ==> cannot be expressed as a product of 3 * k, for some integer k --> not possible
Insufficient

On combining both statements we have.
p>q and q>1 ==> 1<q<p
As p>q ==> p>q-1
So q-1 cannot be expressed as p*k, for some integer k
Thus, we get a definite NO after combining both the statements.
Answer: C

My answer was D
Because the question is asking "Always"
I can be wrong

In order to say a statement is sufficient to answer a given question in the Yes/No type of DS question, that statement has to give you a definite YES or a definite NO. Yes, the statement is sufficient if it gives you a definite NO. But a statement is not sufficient if it gives you YES sometimes (to some values) and NO to some values. In this question, as you have also mentioned "always", each statement is giving YES and NO if you consider them individually. That is not "always". So D cannot be answer to this question as per my knowledge. If you solved in some other way and got definite YES or NO to each statement when considering individually, please share. May be I have committed some mistake while solving. After all, we learn more as we discuss.