Modular Arithmetic

One of the most important tools in number theory is modular arithmetic, which involves the congruence relation. A congruence is the relationship between two numbers, that divided by a third number - called the modulus of the congruence relation - have the same remander. For example, \(32\) is congruent with \(8\) modulus \(12\) since \(32 = 2 \times 12 + 8\) and \(8 = 0 \times 12 + 8\). This relation is usually represented in the following way: \[32=8\,(\mbox{mod }12)\]

In several situations it might be helpful to ignore the multiples of a given number in computations. Think e.g. about the days of the week or the time of the day; in the first case we ignore multiples of \(7\), in the latter, multiples of \(24\) (or multiples of \(12\)). These are examples of "modular arithmetic of modulus \(n\)".

The "12-hour clock arithmetic" is an example of modular arithmetic with modulus \(n=12\). If the time is 7:00am now, then 10 hours later it will be 5:00pm. Usual addition would suggest that the later time should be 7 + 10 = 17, but this is not the answer because clock time "wraps around" every 12 hours (so \(7 + 10\) is equal to \(5\) modulus \(12\)); in 12-hour time, there is no "17 o'clock". Likewise, if the clock starts at 7:00am and 88 hours elapse, then the time will be 11:00 \((7 + 88\) is equal to \(11\) modulus \(12).\) three days latter \((88=3\times 24+16)\). Modular arithmetic is the extension to any \(n\) of the 12-hour clock arithmetic.

Try your ability with clock arithmetic in this clock:

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There are several operations that can be carried out in this arithmetic, like addition and multiplication. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801.