Remainder Theorem

[(a*b*c*……) +(x*y*z*…….)] mod D = [{(a mod D)*(b mod D)*(c mod D)*….} +{x mod D)*(y mod D)*(z mod D)*….}] mod D.

Of all n consecutive numbers in [a, a+n-1], only one is divisible by n.

If x, y, n are three integers, xy – yx = 9*n. Ex: 54-45=9 (x=5, y=4); 93-39=54=9*6.

(m + n)!  mod (m! n!) = 0.

(a^n) mod (a+1) = a if n is odd
                            = 1 if n is even.

For all n being a natural number, n^3 – n is divisible by 6.

10^n – 1 is always divisible by 3 or 9.

If x=aD+b and y=cD+d, (x+y) mod D = (b+d) – D.

If two numbers divided by same divisor leaves the same reminder, then the difference of the two numbers is divisible by the same divisor.

If a number n is divided by x, y, z and it leaves a, b, c as remainders, then n/(x*y*z) will leave a remainder of a + b*x + c*x*y.