Discussions

Hello everyone, I have a question. During a congruence calculation, I found a very strange pattern. Let a,t be integers and d the largest factor of t having a power greater than or equal to 2, If a^2 = 0 (mod t) then a = 0 (mod t/d) For example : a^2 = 0 (mod 8) a^2 = 0 (mod 16) 8 = 2^3 Here 2 is the largest factor 16 = 2^4 = 4^2 Here 4 is the largest factor a = 0 (mod 8/2) = 0 (mod 4) a = 0 (mod 16/4) = 0 (mod 4) Is that true? And if it is, how can we prove it? Honestly, I really don't know what to make of that.

Going to make a gold farm in Minecraft, I would like to know what is the the optimal amount of furnaces to use to smelt the golden swords (not stackable) in order to, not to create a backlog, minimise the amount of fuel needed per hour, and minimise wasted fuel per hour. I have got an answer myself, but I would like to see what your answers are, and also I'm not convinced/confident that I got the right answer. The rates you need to know: 1376.1 swords per hour 1 sword takes 10 seconds to smelt 1 piece of fuel can smelt 12 swords