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For example, I do not have the fact that if the last 2 digits in a number are divisible by 4, that number is then divisible by 4.

I'm pretty sure my teacher will not expect me to have this memorized, and my course definitively doesn't need me to be able to prove it... but I want to be able to.

At what point will be able to do this.. currently I am reading basic mathematics by lang. How many more books would I have to read? Can anyone recommend me a book after langs basic math book? I was thinking of Spivak calculus but idk man. I just find it depressing that I'm not able to prove this right now.