L.C.M. and H.C.F

Euclidean Algorithm : Let h be the H.C.F. of two positive integers x and y, then there exists unique integers m and n such that h = mx + ny.

H.C.F. mod L.C.M. = 0

For two numbers a and b, a*b = H.C.F. (a, b)*L.C.M.(a, b).

If a=b*t + r, then G.C.D. (a, b) = G.C.D. (b, r). 

Largest number of n digits which when divided by x, y, z leaves no remainder. =
n-th digit greatest number – remainder[n-th digit greatest number/L.C.M.(x ,y, z)]

Largest number of n digits which when divided by x, y, z leaves remainder k =
n-th digit greatest number – remainder[n-th digit greatest number/L.C.M.(x, y, z)] + k

Smallest number of n digits which when divided by x, y, z leaves no remainder =
n-th digit greatest number – remainder[n-th digit greatest number/L.C.M.(x, y, z)]

Smallest number of n digits which when divided by x, y, z leaves remainder k =
n-th digit greatest number – remainder[n-th digit greatest number/L.C.M.(x, y, z)] + k

Largest number which when divides x, y, z leaves remainder r in each case = H.C.F. (x-r, y-r, z-r)

Largest number which when divided by x, y, z leaves remainder r in each case = H.C.F. (x-r, y-r, z-r)

Largest number which when divided by x, y, z to leave same remainder in each case= H.C.F. (y-x, z-y, z-x)

Largest number which when divided by x, y, z leave remainders a, b, c in each case= H.C.F. (x-a, y-b, z-c)

Least number which when divided by x, y, z leaves remainder r in each case= L.C.M. (x, y, z) + r