If x and z are positive integers, is x^2 - z^2 odd?

Great question!!

Target question: Is x² - z² odd?

Given: x and z are positive integers

IMPORTANT RULES;
Rule #1) (ODD)² = ODD
Rule #2) (EVEN)² = EVEN
Rule #3) ODD + EVEN = ODD
Rule #4) EVEN + ODD = ODD
Rule #5) ODD - EVEN = ODD
Rule #6) EVEN - ODD = ODD

Statement 1: x + z is odd
By Rule #3 or Rule #4, we can conclude that one number (x or z) is ODD and the other number is EVEN.
So, there are two possible cases:
case a: x is ODD and z is EVEN
case b) x is EVEN and z is ODD

Let's examine each case.
Case a: if x is ODD and z is EVEN, then x² is ODD and z² is EVEN. In this case, x² - z² = ODD - EVEN = ODD
Case b: if x is EVEN and z is ODD, then x² is EVEN and z² is ODD . In this case, x² - z² = EVEN - ODD = ODD
In either case, x² - z² is ODD
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: x - z is odd
By Rule #5 or Rule #6, we can conclude that one number (x or z) is ODD and the other number is EVEN.
So, there are two possible cases:
case a: x is ODD and z is EVEN
case b) x is EVEN and z is ODD

Let's examine each case.
Case a: if x is ODD and z is EVEN, then x² is ODD and z² is EVEN. In this case, x² - z² = ODD - EVEN = ODD
Case b: if x is EVEN and z is ODD, then x² is EVEN and z² is ODD . In this case, x² - z² = EVEN - ODD = ODD
In either case, x² - z² is ODD
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: D