108 6. Integers, Divisors, and Primes
All this is nothing new if we think of “Monday” as 1, “Tuesday” as 2,
etc., and realize that since day 8 is Monday again, we have to replace the
result of any arithmetic operation by its remainder modulo 7. All the above
identities express congruence relations, and are immediate from the basic
properties of congruences.
What about division? In some cases, this is obvious. For example, what
is Sa/We? Translating to integers, this is 6/3, which is 2, i.e., Tu. Check:
Tu·We = Sa.
But what is Tu/We? In our more familiar number systems, this would
be 2/3, which is not an integer; in fact, rational numbers were introduced
precisely so that we could talk about the result of all divisions (except
divisions by 0). Do we have to introduce “fractional days of the week”?
It turns out that this new number system (with only 7 “numbers”) is
nicer! What does Tu/We mean? It is a “number”Xsuch thatX·We = Tu.
But it is easy to check that We·We = Tu; so we have (or at least it seems
to make sense to say that we have) that Tu/We = We.
This gives an example showing that we may be able to carry out division
without introducing new “numbers” (or new days of the week), but can
we always carry out the division? To see how this works, let’s take another
division: We/Fr, and let’s trynotto guess the result; instead, call itXand
show that one of the days of the week must be appropriate forX.
So letX=We/Fr. This means thatX·Fr = We. For each dayXof the
week, the productX·Fr is some day of the week.
The main claim is thatfor different daysX, the productsX·Frare all
different.Indeed, suppose that
X·Fr =Y·Fr.Then
(X−Y)·Fr = Su (6.2)(we used here the distributive law and the fact that Sunday acts like 0).
Now, Sunday is analogous to 0 also in the sense that just as the product
of two nonzero numbers is nonzero, the product of two non-Sunday days is
non-Sunday. (Check!) So we must haveX−Y= Su, orX=Y+Su=Y.
So the daysX·Fr are all different, and there are seven of them, so every
day of the week must occur in this form. In particular, “We” will occur.
This argument works for any division, except when we try to divide by
Sunday; we already know that Sunday acts like 0, and so Sunday multiplied
by any day is Sunday, so we cannot divide any other day by Sunday (and
the result of Su/Su is not well defined; it could be any day).
Congruences introduced in Section 6.7 provide an often very convenient
way to handle these strange numbers. For example, we can write (6.2) in
the form
(x−y)· 5 ≡0 (mod 7)