Answer:
Even
Summary explanation:
If even one factor of the product is even the whole product is even.
If the first two factors are not even, then m+n is even. This is because (2k+1)+(2j+1)==2k+2j+2=2(k+j+1).
Example 3+5=8.
Step-by-step explanation:
Let's do some trials.
Both odd; choose m=3 and n=5
3(5)(3+5)
15(8)
120 is even.
Both even; choose m=2 and n=4
2(4)(2+4)
8(6)
48 is even
One even and one odd; m=2 and n=3
2(3)(2+3)
(6)(5)
30 is even
We could reverse the the choices here but the result is the same.
Appears to be even.
Proof:
Case 1) both odd integers:
Let m=2k+1 and n=2j+1
(2k+1)(2j+1)(2k+1+2j+1)
(2k+1)(2j+1)(2k+2+2j)
2(2k+1)(2j+1)(k+1+j) is even because of the factor of 2 in the product.
Case 2) both even integers
Let m=2k and n=2j.
2k×2j(2k+2j)
Without any work, we can tell this even because of the factor of 2 in the product.
Case 3) one odd and the other even.
If one factor is even, the whole thing is even.
mn(m+n) is always even.
End of proof.
