learn something new everyday

1) Is 0.1212121212...where the 12s are assumed to continue forever a rational number?

solution: let x = 0.1212121212...
so
100x = 12.12121212...
so
100x-x = 12
99x = 12
x = 12\99 QED

2) If day T is the day of the week today and day N is the day of the week in N days, then
day N = (day T + N)mod7
where Sunday = 0, Monday = 1...

3) If a and b are any integers with b non-zero, q,r are non-negative integers such that
a = bq + r
then
gcd(a,b) = gcd(b,r)
And more importantly, this is the basis for the Euclidean algorithm, a method for finding the GCD of two integers.
Its proof is first showing gcd(a,b) <= gcd(b,r) and then show the other way later. Thus the equality.