Modular arithmetic is quite a useful tool in number theory. In particular, it can be used to obtain information about the solutions (or lack thereof) of a specific equation.
In mathematics, modular arithmetic (sometimes called clock arithmetic) is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801.
Time-keeping on this clock uses arithmetic modulo 12. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Usual addition would suggest that the later time should be 7 + 8 = 15, but this is not the answer because clock time "wraps around" every 12 hours; in 12-hour time, there is no "15 o'clock". Likewise, if the clock starts at 12:00 (noon) and 21 hours elapse, then the time will be 9:00 the next day, rather than 33:00. Since the hour number starts over after it reaches 12, this is arithmetic modulo 12. 12 is congruent not only to 12 itself, but also to 0, so the time called "12:00" could also be called "0:00", since 0 ≡ 12 mod 12.
