When we divide a by n using Euclidean division, we obtain a remainder. This remainder can take only a limited range of values: there are just n of them. This new perspective can simplify many calculations and cast a whole host of problems in a different light!
A curious child might wonder why their parents give them an "afternoon snack" at around 4 p.m. And why, four hours after ten o'clock, it is 10 p.m. These are strange additions indeed… The idea behind all this is that when the clock strikes noon, we can start counting the hours again from zero. We can therefore treat "5 o'clock" and "17:00" as equivalent: the difference between these two times is twelve hours. In other words, 5 and 17 have the same remainder upon Euclidean division by 12.
Why restrict ourselves to clocks whose faces are divided into twelve sectors? What would happen with ten-hour clocks? Or thirty-seven-hour clocks? Mischievous and imaginative, mathematicians cannot resist generalizing the idea. This is how congruences arise.
A new arithmetic
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Let a and b be integers, and let n be a natural number at least 2. We say that a and b are congruent modulo n if they have the same remainder upon Euclidean division by n. Equivalently, their difference a – b is a multiple of n. This is written a≡b (mod n). For example: 9 ≡ 24 (mod 5).
