Finding out smallest number which leaves specific remainders when divided with specific divisors.

Type 1: Find smallest number other than k, that leaves remainder k when divided by x, y, z.
How to Solve: take L.C.M of x, y, z and add k to it.

Type 2: Find smallest number which leaves the remainders x-r, y-r, z-r when divided by x, y, z.
How to Solve: take L.C.M of x, y, z and subtract r from it.

Type 3: Find the smallest number which when successively divided by x, y, z leaves remainders a, b, c respectively.
How to Solve: Start from the last divisor, here the last divisor is z which leaves remainder c, smallest number satisfying this condition will be ‘c’.
Now this ‘c’ must have came after the number was divided by y, so the number must have been c*y + b.
Continue in the same manner.

Variations Of The Basic Types Discussed.

(1) Smallest number that leaves remainders x, y, z when divided by a, b, c and leaves remainder X when divided by Y.

This and similar variations are based upon a simple fact.
To a number if we add the l.c.m. of its divisors, the corresponding remainders don’t change.
What this effectively means is we will first find out a no. which leaves remainders x, y, z when divided by a, b, c respectively and then keep adding the L.C.M of a, b, c to it until the other condition(s) are not satisfied.


Courtesy : Varied sources on Internet.

P.S-> For those of you, who haven't been using these methods, please try them on at-least 5 problems of each type before relying on them for the D-day.