Difference between revisions of "User:Catsmeow12"
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| − | + | Greetings! I am CatsMeow12. I like math, cats, coding, animation, and various other stuff. If you see this, you've either clicked the [https://artofproblemsolving.com/wiki/index.php/Special:Random random page] button or just like to explore. In any case, you have landed here. | |
| − | I like math, | + | =Math Content= |
| + | This is an AoPS Wiki, so I assume that at least ''some'' math is expected here. So, I will post some things I like or came up with. | ||
| + | |||
| + | ==Modular Division== | ||
| + | It is generally accepted that division is not defined in modular arithmetic. An example like <math>6n\equiv3\pmod{9}</math> is usually presented, and both "sides" of the congruence are usually divided by their GCF. Then the divided congruence usually becomes something like <math>2n\equiv1\pmod{9},</math> which is simplified to something like <math>n\equiv5\pmod{9}.</math> At this point, it is usually shown that the attained congruence is a solution for the original congruence but doesn't account for all the possible solutions. So, the conclusion is reached that division doesn't exist in modular arithmetic. Some are more careful with their wording and state that division is not '''defined''' in modular arithmetic. | ||
| + | |||
| + | Notice, though, that you can't solve the above equation without dividing both sides '''and''' the modulus, even with modular inverses. | ||
Latest revision as of 18:11, 27 September 2023
Greetings! I am CatsMeow12. I like math, cats, coding, animation, and various other stuff. If you see this, you've either clicked the random page button or just like to explore. In any case, you have landed here.
Math Content
This is an AoPS Wiki, so I assume that at least some math is expected here. So, I will post some things I like or came up with.
Modular Division
It is generally accepted that division is not defined in modular arithmetic. An example like 
is usually presented, and both "sides" of the congruence are usually divided by their GCF. Then the divided congruence usually becomes something like 
which is simplified to something like 
At this point, it is usually shown that the attained congruence is a solution for the original congruence but doesn't account for all the possible solutions. So, the conclusion is reached that division doesn't exist in modular arithmetic. Some are more careful with their wording and state that division is not defined in modular arithmetic.
Notice, though, that you can't solve the above equation without dividing both sides and the modulus, even with modular inverses.
