Modular Arithmetic Applet

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Instructions

Easy Peasy

Input values into the two blue boxes, then click "Answer" to draw the clock wedges and give the answer. As the notes above say, if you input large values, things might start to become slightly unclear, but it's handy for getting to know the system using 1 and 2 digit numbers.

Background on Modular Arithmetic

This tool can be used to help visualise modular arithmetic; a system where numbers "wrap around" to \(0\) when they reach the modulus value. For example, counting in \(\bmod{4}\) would go \(1, 2, 3, 0, 1, 2, ...\). Values are said to be congruent if they have the same value with respect to a certain modulus. For example, \(15 \equiv 3 \bmod{12}\) is something you might be familiar with, because 24hour time uses modular arithmetic; 15:00 is 3:00pm, the time "wraps around" at 12 and we start counting again. This is why the system is sometimes called clock arithmetic. You don't only have to use modulo 12 though, the default above is modulo 6, but change it to any 2 digit number you want!

A side note

You may be wondering why we can't say \[15 \equiv 27 \bmod{12} \\[6pt] 15 - 12 = 27 - 12 - 12 = 3\] using what we learnt above. Well we can! in fact, there are infinitely many congruent numbers to any other number, but for simplicty the answers above assume that you want the answer, which can be seen as a remainder, smaller than the modulus itself.