Answer:
The expression [tex]m^3-n^3[/tex] is even if both variables (m and n) are even or both are odd
Step-by-step explanation:
Let's remember the logical operations with even and odd numbers
odd*odd=odd
even*even=even
odd*even=even
odd-odd=even
even-even=even
even-odd=odd
Now, the original expression is:
[tex]m^3-n^3[/tex] which can be expressed as:
[tex](m*(m*m))-(n*(n*n))[/tex]
If m and n are both odd, then:
[tex](m*(m*m))=odd*(odd*odd)=odd*(odd)=odd[/tex]
[tex](n*(n*n))=odd*(odd*odd)=odd*(odd)=odd[/tex]
Then, [tex](m*(m*m))-(n*(n*n))=odd-odd=even[/tex]
If m and n are both even, then:
[tex](m*(m*m))=even*(even*even)=odd*(even)=even[/tex]
[tex](m*(m*m))=even*(even*even)=odd*(even)=even[/tex]
Then, [tex](m*(m*m))-(n*(n*n))=even-even=even[/tex]
Finally if one of them is even, for example m, and the other is odd, for example n, then:
[tex](m*(m*m))=even*(even*even)=odd*(even)=even[/tex]
[tex](n*(n*n))=odd*(odd*odd)=odd*(odd)=odd[/tex]
Then, [tex](m*(m*m))-(n*(n*n))=even-odd=odd[/tex]
In conclusion, the expression [tex]m^3-n^3[/tex] is even if both variables (m and n) are even or both are odd. If one of them is even and the other one is odd, then the expression is odd.
