In the comments over the past few weeks, and based on some of the issues raised recently, the concept of congruence comes up often.

In number theory (there is also geometric congruence), two integers are said to be congruent when they give the same remainder when divided by a third, called the modulus. Thus, 7 and 19 are congruent with respect to 4 because both, when divided by 4, give a remainder of 3.

Some congruences are obvious; for example, all odd numbers are congruent with respect to 2, since they all give 1 as a remainder when divided by 2 (what can we say, in this sense, about numbers ending in 1?).

The congruence relationship is expressed by three parallel lines and the module in parentheses:

a ≡ b (mod m)

means that a and b are congruent with respect to m.

Congruence can also be defined as the relationship between two integers whose difference is divisible by a third. If a and b are congruent with respect to m, they give the same remainder, r, when divided by m, whence:

a = pm + r

b = qm + r

where p and q are integers, and therefore:

a – b = (p – q)m

then a – b is divisible by m.

Congruence is the basis of modular arithmetic, introduced by Gauss at the beginning of the 19th century with his book *Disquisitions Arithmeticae*. And modular arithmetic is also known as “clock arithmetic”, because clocks very graphically illustrate the equivalence relationship of hours with respect to module 12: thus, 7 and 19 hours are represented on conventional clocks of the same way: with the major hand at 12 and the minor at 7.

### problematic watches

One cannot talk about clock arithmetic without thinking about the numerous problems and riddles (some well known and others not, some easy and others not so much) that have clocks as protagonists. They constitute a whole section of ingenuity problems, which in turn can be divided into three subsections: needle clocks, hourglasses and digital clocks. Let’s look at some of the first type:

A striking clock, of those that chime the hours, takes 6 seconds to strike 6. How long will it take to strike 12?

Near my house there are two clocks that strike the hours at different speeds: one strikes three times at the same time as the other strikes two. They are synchronized and start ringing at the same time. At what time does the slow clock strike two more chimes when the fast one has stopped chiming? (Based on true events, like the last one).

At 12 o’clock, the three hands of the clock —the hour hand, the minute hand, and the second hand— coincide exactly (they present arms to the Sun, as Ramón Gómez de la Serna would say). When will the three meet again?

And as a climax, a well-known classic, but an obligatory mention in this context. Classic and historical, because the anecdote is real:

One afternoon, Kant saw that the clock in his house had stopped. Shortly after, he was walking to visit a friend, at whose house he noticed the time on a wall clock. After conversing for a long time with his friend, Kant returned to his house by the same path, walking, as usual, with the steady and regular gait that he had not changed for twenty years. He had no idea how long it had taken to make the way back, since his friend had recently moved and Kant had not yet timed the journey. However, as soon as he got home he set the clock on time. How did he do it?

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#arithmetic #clock