Given that p and q are different prime numbers and a, b, and c are consecutive integers such that a < b < c and we need to find which of the following must be odd

Since p and q are different prime numbers
So, p can be 2 (Even) or any other odd prime number
q can be 2(if p is not 2) or any other odd prime number apart from what p is
=> Either of p or q can be even or both can be odd too.

a, b, c are consecutive integers such that a < b < c
=> if a is odd then b is even and c is odd
=> if a is even then b is odd and c is even
=> a and c will be either both odd or both even, and b will be even or odd accordingly

(A) ap
Now, p can be even => ap can be even => FALSE

(B) bp
Now, p can be even => bp can be even => FALSE

(C) pq
Now, p can be even => pq can be even => FALSE

(D) a + b
Now, a can be odd, making b as even and we know odd + Even = odd
a can be even, making b as odd and we know even + odd = odd
=> a + b will ALWAYS be odd
=> TRUE

(E) a + q
Now, both a and q can be odd or both can be even making a + q = even => FALSE

So, Answer will be D
Hope it helps!

Watch the following video to learn How to Sequence problems