<2> if we take for granted that every number can be decomposed as product of primes in an unique way - except the order:
m divides (a-b) hence a-b = m*q
n divides (a-b) hence all the prime factors in n are either in m or in q.
gcd(m,n)=1, so no factor of n is in m. Then all factors of n are in q, or q=n*r for some other integer r. Replacing in the first equality: a-b = m*n*r, proving a = b (mod m*n)
<3> x^2-1 = (0 mod p), or (x-1)*(x+1) = pq for some integer q. Since p is prime it must appear in at least one of the decompositions: of (x-1) or of (x+1). Hence p divides (x-1) or (x+1)
<4>
For each 365 day year there are 7 possible calendars, depending on which of the 7 days of the week falls on January 1st. For leap years - 366 days - the same argument applies, yielding 7 more combinations. These later combination are different from any of the first 7 because if January 1st happens to be the same, March 1st is necessarily one day later. So there are 14 distinct calendars.
To check that all combinatios will be - were already - realised, take periods of 4 years each starting January 1st. In any such period there are always 1461 days, which decomposes as 7*209 - 2. Hence if the January 1st of a period is Monday - for example - 4 years later January 1st will be Saturday, exactly two week days behind. Cycling through more 4 year periods we can find years that start with any given day of the week; this is known because 7 is a prime number, but in this case it is easy to convince oneself that it is true, through a method similar to counting on one's fingers:
Monday>Saturday>Thursday>Tuesday>Sunday>Friday>Wednesday
We also notice that we only need 7 fingers, so 7 periods of 4 years are enough to find all possible 7 calendars for non-leap years. For leap years it suffices to shift the eriods in consideration so that they start with a leap year.
Finally, the Gregorian calendar makes exception to the rule that every fourth year is a leap year, but at most once 100 years. As shown above, we can find all 14 calendar kinds within any 31 years period inside a 100 years where there are no exceptions.
