The correct statement among the given options is iii. p^2-q^2 is even.
Let's analyze each statement:
i. pq is even:
If both p and q are greater than 2 and primes, they are odd numbers. When you multiply two odd numbers, the result is always odd. So, pq must be odd, not even.
ii. pq is odd:
As discussed above, pq must indeed be odd because the product of two odd numbers is always odd.
iii. p^2 −q^2 is even:
Let's consider p^2 −q^2 .
This can be rewritten using the difference of squares formula:
p^2 −q^2 =(p+q)(p−q).
We know both p and q are odd primes greater than 2. So, p+q will be even because the sum of two odd numbers is even. Similarly,
p−q will also be even since subtracting an odd number from an odd number results in an even number.
Therefore, their product, (p+q)(p−q), will be even as the product of two even numbers is always even.
So, the correct statement among the given options is iii. p^2-q^2 is even.
Question
If p and q are primes greater than 2 , which of the following must be true ? i. p q is even
ii. pq is odd
iii. p^2-q^2 is even