If p and q are positive integers. If p is divided by 2, the remainder

Both p and q have to be ODD if they leave a remainder of 1 when p is divided by 2 and q is divided by 6

Odd * Odd number = Odd number(always) and any odd number added to 1 is even.
Hence, I is always true.

Also Odd/2 is never going to be an integer. Hence II is false

In the case when both p = q = 1, pq is a multiple of 12,
but if p = 1, q = 7, pq is not a multiple of 12. Hence III is also false(Option A)

We are given that p and q are positive integers. If p is divided by 2, the remainder is 1, and if q is divided by 6, the remainder is 1. Thus, we have:

p/2 = Q + 1/2

p = 2Q + 1

Thus, p can be values such as 1, 3, 5, ..., which are all odd values.

Let’s study q in a similar fashion.

q/6 = Q + 1/6

q = 6Q + 1

Thus, q can be values such as 1, 7, 13, ..., which are all odd values.

Let’s analyze the Roman numerals:

I. pq + 1 is even.

Since p and q are odd numbers, the product pq is odd, and thus pq + 1 must be even. So I is true.

II. pq/2 is an integer.

Again, since the product pq is odd, pq/2 will not be an integer. So II is not true.

III. pq is a multiple of 12.

Once again, since the product pq is odd and 12 is even, pq/12 will not be an integer. So III is not true.

Answer: A

it is given that both p and q are odd

hence pq + 1 is even is true (I)
pq/2 is not an integer as odd/even is decimal (II)

if p and q are 1 and 7 - pq is not a multiple of 12 - False

Only I MUST be true - hence answer A