Prove that 11x - 7 is even if and only if x is odd

Prove that x - 7 is even if and only if x is odd. This is a good example of an if and only if proof. I hope this helps.

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[martinhawrylkiewicz]

Nice proof. You can also prove it directly the 2nd part which to me is much simpler I think. If 11x - 7 is even then 11x - 7 = 2n for some n in Z, but then 11x = 2n + 7 = 2(n + 3) + 1 for some integer n + 3 which means 11x is odd together with 11 being odd, means x is odd.

[andrewzhang]

It's easier just to do 11x-7 = 0 (mod 2) if and only if x=1 mod 2

[CoolAid]

Math sorcerer you are my idol. I want to be a mathematician like you. I want to be a math professor

[peter-ckdb]

Why did yt recommend me this at 5am???

[lolzhunter]

this one is actually quite easy relatively speaking, even*odd = even and odd*odd is odd, even-odd is odd and odd-odd is even, using these 4 rules you can quickly show that the only way for the end result to be even is if x is odd
for example if x was even then that would be 11*even, and 11 is odd so odd*even, and that by definition gets you an even number, you then minus 7 which is odd which means even-odd which is odd, which shows that when x is even, you always get odd

when x is odd you have 11 (which is odd) times an odd number, so odd*odd which gets you odd, then you minus 7 which we know is odd and you get odd-odd which is always even, therefore when x is odd you get even

EDIT:
since this proof is so general this also inadvertently proves that this rule holds for any equation in the format of odd*x±odd, where ± is plus or minus since the rules for odd even minus are the same for plus, so (2n-1)*x±(2m-1), n and m being any number you want since the result will be odd, getting the original equation the values would be n=6 and m=4

[darly]

Why did I get recommended this? I strictly watch kawaii vtubers and leftist political content. Good shit though dude