Discrete Mathematics: Elementary and Beyond

6.8 Strange Numbers 107

6.7.3How would you definea≡b(mod 0)?

6.7.4(a) Find two integersaandbsuch that 2a≡ 2 b(mod 6), buta≡
b(mod 6). (b) Show that ifc = 0 andac≡bc(modmc), thena≡b(modm).

6.7.5Letpbe a prime. Show that ifx, y, u, vare integers such that x≡
y(modp),u, v >0, andu≡y(modp−1), thenxu≡yv(modp).

6.8 Strange Numbers........................

What is Thursday + Friday?
If you don’t understand the question, ask a child. He/she will tell you
that it is Tuesday. (There may be some discussion as to whether the week
starts with Monday or Sunday; but even if we feel it starts with Sunday,
we can still say that Sunday is day 0.)
Now we should not have difficulty figuring out that Wednesday·
Tuesday = Saturday, Thursday^2 = Tuesday, Monday−Saturday =
Tuesday, etc.
This way we can do arithmetic operations with the days of the week:
We have introduced a new number system! In this system there are only 7
numbers, which we call Su, Mo, Tu, We, Th, Fr, Sa, and we can carry out
addition, subtraction, and multiplication just as with numbers (we could
call them Sleepy, Dopey, Happy, Sneezy, Grumpy, Doc, and Bashful; what
is important is how the arithmetic operations work).
Not only can we define these operations; they work pretty much like
operations with integers. Addition and multiplication are commutative,

Tu + Fr = Fr + Tu, Tu·Fr = Fr·Tu,

and associative,

(Mo+We)+Fr=Mo+(We+Fr), (Mo·We)·Fr=Mo·(We·Fr),

and distributive,

(Mo + We)·Fr = (Mo·Fr) + (We·Fr).

Subtraction is the inverse of addition:

(Mo + We)−We = Mo.

Sunday acts like 0:

We+Su=We, We·Su = Su,

and Monday acts like 1:
We·Mo = We.