AlanH
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 Jan3 comment prove that $\text{rank}(AB)\ge\text{rank}(A)+\text{rank}(B)-n.$ Have you tried working out an example? Try a few by hand and see if you can generalize it. Or you can try showing it by contradiction; though I always feel direct proofs are much more clear. Dec20 comment If an integer is divisible by 8 and 15, then the integer also must be divisible by which of the following? Ah! I figured it out just now. Thank you Dec20 comment If an integer is divisible by 8 and 15, then the integer also must be divisible by which of the following? What else does it imply? (I actually thought about it, but I'm quite hesitant to state what I think). Dec20 comment Pigeon-hole Principle: Does this proof have a typo? Ah yes, that's what I figured. I was thinking that if it were less than, say a radius of 0.01, then you could definitely arrange the dots so that the circumcircles don't contain them. Dec9 comment I can't figure out this combinatorics problem… Or at least why my solution doesn't work. @JoshKeneda Why is it that (1/10)*(1/9) can't represent 8, THEN 1? It seems like if I wanted to pick an 8, and then a 1, the probability would be the same. Dec9 comment I can't figure out this combinatorics problem… Or at least why my solution doesn't work. I'm not sure I understand your question, but in my solution, it would be for 1 case. The second and third cases would be 2,x1,x2,9 and 3,y1,y2,10, where x1,x2 are numbers between 2 and 9, and y1,y2 are numbers between 3 and 10.